Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C32 | C22⋊C4 |
| commutator | C22 | C22×C32 |
| Frattini, cosocle | C2×C32 | C23 |
| socle | C2 | C2×C22⋊C32 |
| prime radical | C2×#41 | C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
180 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,4,11,5,12,22,41,6,13,23,42,26,44,62,87,7,14,24,43,27,45,63,88,29,46,65,89,68,90,104,120,8,10,25,36,28,37,64,78,30,38,66,79,69,81,105,113,31,39,67,80,70,82,106,114,71,83,107,115,108,116,125,127)(3,15,17,40,18,47,53,84,19,48,54,85,56,91,97,117,20,49,55,86,57,92,98,118,58,93,99,119,100,121,123,128,16,9,21,32,50,33,59,72,51,34,60,73,94,75,101,109,52,35,61,74,95,76,102,110,96,77,103,111,122,112,124,126);(1,3)(2,9)(4,21)(5,18)(6,19)(7,20)(8,16)(10,15)(11,40)(12,33)(13,34)(14,35)(17,25)(22,59)(23,60)(24,61)(26,56)(27,57)(28,50)(29,58)(30,51)(31,52)(32,36)(37,47)(38,48)(39,49)(41,84)(42,85)(43,86)(44,75)(45,76)(46,77)(53,64)(54,66)(55,67)(62,101)(63,102)(65,103)(68,100)(69,94)(70,95)(71,96)(72,78)(73,79)(74,80)(81,91)(82,92)(83,93)(87,117)(88,118)(89,119)(90,112)(97,105)(98,106)(99,107)(104,124)(108,122)(109,113)(110,114)(111,115)(116,121)(120,128)(123,125)(126,127);(3,16)(9,15)(17,21)(18,50)(19,51)(20,52)(32,40)(33,47)(34,48)(35,49)(53,59)(54,60)(55,61)(56,94)(57,95)(58,96)(72,84)(73,85)(74,86)(75,91)(76,92)(77,93)(97,101)(98,102)(99,103)(100,122)(109,117)(110,118)(111,119)(112,121)(123,124)(126,128)Polycyclic presentation:
2^9|a^2=e,e^2=f,f^2=g,g^2=h,h^2=i,b^a=bd,d^a=di,c^b=ci,e^b=ei176 conjugacy classes — view the full character table.
| maximal subgroup of | #72321739, #72321799, #72321810, … |
| maximal normal subgroup of | #72321739, #72321799, #72321810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60626);