Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139572 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1910 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #9468 |
| prime radical | #24285 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
499 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,13)(4,19)(5,23,26,9)(6,28)(7,12)(8,16)(10,14)(11,20)(15,41)(17,37,24,39)(18,51)(21,55,25,52)(22,53)(27,34,54,56)(29,49,30,47)(31,60)(32,45)(33,48,35,43)(36,44)(38,42)(40,46)(50,62,57,61)(58,63,59,64);(1,3,14,42,6,18,22,45,7,15,44,60,20,46,4,16)(2,8,19,40,11,31,36,41,12,32,53,51,28,38,10,13)(5,24,43,64,27,57,25,47,26,17,48,63,54,50,21,49)(9,30,52,62,34,59,33,37,23,29,55,61,56,58,35,39);(2,9)(3,15)(8,29)(10,35)(11,34)(12,23)(13,37)(16,45)(17,24)(18,46)(19,52)(28,56)(30,32)(31,58)(33,36)(38,59)(39,41)(40,61)(42,60)(47,49)(50,57)(51,62)(53,55)(63,64)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,f^c=fh,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72281154, #72342931, #72343002, … |
| maximal normal subgroup of | #72281154, #72342931, #72343002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46768);