Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1812 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #8609 |
| prime radical | #525 | D4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #2113 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
295 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 84 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)(5,23,24,9)(6,26)(7,12)(8,16)(10,14)(11,20)(15,38)(17,34,43,36)(18,47)(21,51,52,48)(22,49)(25,32,50,54)(27,55,28,45)(29,58)(30,42)(31,40,53,62)(33,41)(35,39)(37,44)(46,60,63,59)(56,61,57,64);(1,3,14,39,6,18,22,42,7,15,41,58,20,44,4,16)(2,8,19,37,11,29,33,38,12,30,49,47,26,35,10,13)(5,17,40,61,25,46,52,55,24,43,62,64,50,63,21,45)(9,28,48,60,32,57,53,34,23,27,51,59,54,56,31,36);(2,9)(3,15)(8,27)(10,31)(11,32)(12,23)(13,34)(16,42)(17,43)(18,44)(19,48)(26,54)(28,30)(29,56)(33,53)(35,57)(36,38)(37,59)(39,58)(45,55)(46,63)(47,60)(49,51)(61,64)Polycyclic presentation:
2^9|a^2=f,b^2=dg,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,f^c=fh,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72287344, #72287357, #72287362, … |
| maximal normal subgroup of | #72287344, #72287357, #72287362, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41970);