Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×D16 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C2 | C2×#1331 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
249 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,5,11,7,12,27,40,8,14,28,41,31,13,6,29)(3,15,18,44,19,45,50,63,16,43,47,64,20,51,30,36)(4,21,23,37,25,52,58,39,26,54,59,38,60,53,24,10)(9,22,33,48,35,56,62,46,32,55,61,49,42,57,34,17);(1,3)(2,9)(4,22)(5,18)(6,30)(7,19)(8,16)(10,36)(11,33)(12,35)(13,42)(14,32)(15,21)(17,24)(20,31)(23,48)(25,56)(26,55)(27,50)(28,47)(29,34)(37,44)(38,64)(39,63)(40,62)(41,61)(43,54)(45,52)(46,58)(49,59)(51,53)(57,60);(3,16)(9,32)(15,43)(17,46)(18,47)(19,20)(22,55)(30,50)(33,61)(34,62)(35,42)(36,63)(44,64)(45,51)(48,49)(56,57)Polycyclic presentation:
2^9|a^2=e,d^2=fh,e^2=g,f^2=hi,g^2=i,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=df,f^b=fh,h^b=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72225304, #72225308, #72225312, … |
| maximal normal subgroup of | #72225304, #72225308, #72225312, … |
| maximal quotient of | #72226994 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57263);