Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7414 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1345 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #423 | C23 |
| socle | C2 | C2×#1345 |
| prime radical | C2×#575 | C4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
459 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 459 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,13,26,16,28,15,31)(4,8)(5,19)(6,11)(7,14,23,20,24,30,22,29)(9,21)(10,18)(17,27)(25,32);(1,3)(2,7)(4,13)(5,20)(6,16)(8,22)(9,26)(10,23)(11,24)(12,27)(14,17)(15,18)(19,31)(21,29)(25,28)(30,32);(1,4,6,18)(2,8,11,10)(3,13,16,15)(5,17,21,32)(7,22,24,23)(9,25,19,27)(12,26,28,31)(14,29,30,20)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=gi,e^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57425);