Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#120 |
| prime radical | #537 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,3,4,8)(5,9,13,16,14,11,12,15)(17,18)(19,25)(20,22)(21,29)(23,28)(24,31)(26,30)(27,32);(1,3)(2,7)(4,10)(5,9)(6,8)(11,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,30)(22,27)(24,32)(25,29)(28,31);(1,4,7,6)(2,8,3,10)(5,12,14,13)(9,15,11,16)(17,20,22,18)(19,26,27,23)(21,24,31,29)(25,30,32,28);(1,5,6,13,7,14,4,12)(2,9,10,16,3,11,8,15)(18,20)(19,27)(24,29)(28,30)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=g,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dh,e^b=eh,f^b=fi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116334);