Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,27,22,26,21,25)(19,20,29,32,28,31,24,30);(1,3,5,2)(4,7,8,6)(17,19)(18,20)(21,24)(22,28)(23,29)(25,30)(26,31)(27,32);(1,4,5,8)(2,6,3,7)(10,13)(14,15)(17,20)(18,24)(19,27)(21,30)(22,31)(23,32)(25,28)(26,29);(2,3)(6,7)(9,11,12,16)(10,14,13,15)(17,21,22,23)(18,25,26,27)(19,24,28,29)(20,30,31,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292000);