Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153082 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2218 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13448 |
| prime radical | #33981 | C2 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
254 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2)(3,18)(4,25)(5,33)(6,37)(7,13)(8,43)(9,46)(10,17)(11,20)(12,28)(14,36)(15,34)(16,45)(19,48)(21,62)(22,75)(23,77)(24,60)(26,65)(27,84)(29,58)(30,92)(31,94)(32,80)(35,61)(38,53)(39,100)(40,59)(41,97)(42,96)(44,49)(47,66)(50,102)(51,73)(52,72)(54,101)(55,88)(56,87)(57,91)(63,98)(64,71)(67,107)(68,119)(69,120)(70,110)(74,118)(76,112)(78,85)(79,93)(81,117)(82,123)(83,116)(86,103)(89,105)(90,125)(95,114)(99,111)(104,108)(106,109)(113,127)(115,121)(122,128)(124,126);(1,3)(2,11)(4,26)(5,21)(6,38)(7,20)(8,44)(9,47)(10,24)(12,54)(13,49)(14,60)(15,62)(16,64)(17,53)(18,33)(19,29)(22,34)(23,45)(25,48)(27,68)(28,70)(30,67)(31,95)(32,83)(35,99)(36,74)(37,50)(39,102)(40,71)(41,73)(42,72)(43,51)(46,52)(55,113)(56,116)(57,117)(58,107)(59,112)(61,66)(63,118)(65,84)(69,89)(75,96)(76,98)(77,97)(78,119)(79,120)(80,110)(81,85)(82,93)(86,127)(87,122)(88,101)(90,104)(91,106)(92,108)(94,109)(100,111)(103,114)(105,115)(121,126)(123,125)(124,128);(1,4,5,27,34,85,42,91,9,31,35,86,39,88,6,28,10,32,36,87,98,126,40,89,45,93,41,90,8,30,7,29)(2,12,13,55,43,103,97,94,16,57,59,78,63,84,14,25,17,58,37,92,100,125,61,79,46,105,96,124,15,56,33,80)(3,19,20,67,44,104,73,82,23,69,71,121,76,122,74,83,24,70,38,101,102,127,99,95,47,106,72,81,22,68,21,26)(11,48,18,65,62,119,75,117,52,109,66,114,111,113,50,54,53,110,60,116,118,128,112,115,64,120,77,123,51,108,49,107)Polycyclic presentation:
2^9|c^2=d,d^2=gh,e^2=g,f^2=g,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fghi,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=eghi,f^b=fgh,g^b=gh,h^b=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320516, #72320849, #72320868 |
| maximal normal subgroup of | #72320516, #72320849, #72320868 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60508);