Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150144 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2140 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C2 | #11661 |
| prime radical | #3414 | C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
220 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 220 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:
(1,2,6,13,8,14,39,61,9,16,40,62,43,15,7,41)(3,17,20,65,21,66,78,124,23,67,79,126,22,80,42,56)(4,24,27,83,29,55,28,53,30,54,38,84,94,11,49,52)(5,31,34,58,36,96,108,60,37,98,109,59,90,97,35,12)(10,32,45,75,47,100,114,77,48,102,115,76,63,101,46,19)(18,68,70,123,72,110,87,125,73,117,71,128,25,64,86,119)(26,88,91,51,93,95,92,82,33,103,105,57,107,120,106,121)(44,89,111,74,50,116,112,85,81,104,113,69,118,99,122,127);(1,3)(2,10)(4,25)(5,32)(6,20)(7,42)(8,21)(9,23)(11,50)(12,56)(13,45)(14,47)(15,63)(16,48)(17,31)(18,29)(19,35)(22,43)(24,81)(26,89)(27,86)(28,70)(30,72)(33,104)(34,75)(36,100)(37,102)(38,87)(39,78)(40,79)(41,46)(44,54)(49,71)(51,119)(52,112)(53,122)(55,118)(57,125)(58,65)(59,126)(60,124)(61,114)(62,115)(64,88)(66,96)(67,98)(68,95)(69,105)(73,94)(74,91)(76,109)(77,108)(80,97)(82,123)(83,113)(84,111)(85,92)(90,101)(93,116)(99,107)(103,110)(106,127)(117,120)(121,128);(1,4,8,29,9,30,43,94)(2,11,15,54,16,55,14,24)(3,18,22,25,23,73,21,72)(5,33,36,107,37,26,90,93)(6,38,39,49,40,27,7,28)(10,44,47,50,48,81,63,118)(12,57,59,82,60,51,58,121)(13,53,41,83,62,52,61,84)(17,64,66,68,67,110,80,117)(19,74,76,127,77,69,75,85)(20,71,42,87,79,70,78,86)(31,95,97,88,98,120,96,103)(32,99,101,104,102,116,100,89)(34,91,108,92,109,105,35,106)(45,113,114,122,115,111,46,112)(56,123,65,125,124,128,126,119)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=fh,e^2=g,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=df,f^b=fh,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72320449 |
| maximal normal subgroup of | #72320449 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57340);