Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134747 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
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 | #532 | D4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #2113 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
299 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 299 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)(3,14)(4,20)(5,11,24,9)(6,26)(7,13)(8,16)(10,15)(12,21)(17,38,47,36)(18,49)(19,40)(22,51,54,50)(23,52)(25,32,53,35)(27,45,28,56)(29,58)(30,46)(31,42,33,59)(34,43)(37,41)(39,44)(48,60,63,61)(55,64,57,62);(1,3,15,41,6,18,23,46,7,19,43,58,21,44,4,16)(2,8,20,39,12,29,34,40,13,30,52,49,26,37,10,14)(5,17,42,62,25,48,54,56,24,47,59,64,53,63,22,45)(9,27,50,60,32,55,33,38,11,28,51,61,35,57,31,36);(2,9)(5,24)(8,27)(10,31)(11,13)(12,32)(14,36)(17,47)(20,50)(22,54)(25,53)(26,35)(28,30)(29,55)(33,34)(37,57)(38,40)(39,60)(42,59)(45,56)(48,63)(49,61)(51,52)(62,64)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,e^c=ei,f^c=fhi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287329, #72287337, #72287362, … |
| maximal normal subgroup of | #72287329, #72287337, #72287362, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41971);