Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149759 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11501 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | #407 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #2921 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
405 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 405 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(1,2,5,11,7,8,21,22,6,12,19,29,4,10,18,27)(3,13,15,30,14,25,31,28,16,26,20,32,17,9,23,24);(2,8)(4,7)(5,19)(9,13)(10,12)(11,22)(14,17)(15,20)(24,30)(25,26)(27,29)(28,32);(1,3)(2,9,12,25)(4,17)(5,20)(6,16)(7,14)(8,13,10,26)(11,28,29,24)(15,19)(18,31)(21,23)(22,32,27,30)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=i,e^2=g,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,d^b=di,e^b=efi,g^b=gi,d^c=di,e^c=ei,e^d=eh,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56955);