Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2196 |
| commutator | #124 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| socle | C2 | #12759 |
| prime radical | #61100 | C2 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
807 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 807 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,7,13,5,14,28,43)(3,15,19,34,17,40,53,44)(4,21,24,41,26,55,57,38)(6,12,29,11,27,42,8,31)(9,30,35,32,36,18,46,20)(10,25,37,58,39,22,56,23)(16,47,50,33,52,54,64,62)(45,48,60,59,61,51,63,49);(1,3)(2,9)(4,16)(5,17)(6,19)(7,30)(8,32)(10,33)(11,40)(12,35)(13,44)(14,36)(15,31)(18,28)(20,29)(21,54)(22,51)(23,50)(24,59)(25,48)(26,52)(27,53)(34,43)(37,63)(38,45)(39,62)(41,61)(42,46)(47,55)(49,57)(56,60)(58,64);(1,4)(2,10)(3,16)(5,26)(6,23)(7,24)(8,22)(9,33)(11,41)(12,21)(13,37)(14,39)(15,45)(17,52)(18,49)(19,50)(20,48)(25,29)(27,58)(28,57)(30,59)(31,38)(32,51)(34,60)(35,54)(36,62)(40,61)(42,55)(43,56)(44,63)(46,47)(53,64)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dh,e^b=eghi,f^b=fghi,g^b=gi,d^c=dgi,g^c=gi,h^c=hi,e^d=ehi,f^d=fhi,g^e=gi,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59380);