Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125651 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1655 |
| commutator, Frattini, cosocle | #376 | C23 |
| socle | C2 | #7745 |
| prime radical | #10012 | C2 |
| LCS | G ▷ #376 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #376 ◁ G | |
| LECS | G ▷ #376 ▷ C22×C4 ▷ C2 ▷ C1 | |
388 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(8,24)(9,22)(10,23)(11,28)(14,30)(17,21)(20,25)(26,29)(27,32);(1,2,5,17,6,10,19,9)(3,8,18,25,12,26,13,27)(4,7,16,21,15,23,31,22)(11,29,30,32,28,24,14,20);(1,3)(2,8)(4,14)(5,18)(6,12)(7,20)(9,27)(10,26)(11,16)(13,19)(15,30)(17,25)(21,29)(22,24)(23,32)(28,31)Polycyclic presentation:
2^9|b^2=fgh,d^2=f,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,d^b=df,e^b=egi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,e^d=ei41 conjugacy classes — view the full character table.
| maximal subgroup of | #72217950, C2×#125841, #72279736, … |
| maximal normal subgroup of | #72217950, C2×#125841, #72279736, … |
| maximal quotient of | #72214494, #72214507, #72214515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33037);