Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C8 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #8316 |
| prime radical | #3529 | C4 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
603 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 603 conjugacy classes of subgroups — too many to draw. Only its 110 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:
(1,2,5,10,6,12,25,37,7,11,26,36,27,38,52,60)(3,13,16,39,17,40,44,61,14,32,23,49,41,57,50,63)(4,18,21,35,22,45,48,59,20,9,24,33,47,34,51,58)(8,28,30,42,31,53,56,19,29,43,54,46,55,62,64,15);(1,3)(2,8)(4,19)(5,23)(6,17)(7,14)(9,32)(10,30)(11,29)(12,31)(13,18)(15,20)(16,26)(21,28)(22,46)(24,43)(25,50)(27,41)(33,49)(34,57)(35,39)(36,54)(37,56)(38,55)(40,45)(42,47)(44,52)(48,53)(51,62)(58,63)(59,61)(60,64);(3,14)(8,29)(9,18)(15,19)(16,23)(17,41)(28,43)(30,54)(31,55)(33,35)(34,45)(42,46)(44,50)(53,62)(56,64)(58,59)Polycyclic presentation:
2^9|a^2=f,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=di,e^b=egi,f^b=fgi,d^c=dg,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239956, #72250202, #72250207, … |
| maximal normal subgroup of | #72239956, #72250202, #72250207, … |
| maximal quotient of | #72239963 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36812);