Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,26)(20,23)(21,24)(22,27)(25,29)(28,31)(30,32);(5,7,8,6)(10,14)(15,16)(17,19)(18,22)(20,27)(21,28)(23,26)(24,30)(25,31)(29,32);(1,2,3,4)(9,11,13,12)(10,15,14,16)(18,23)(19,27)(24,25)(28,32);(9,12,13,11)(10,16,14,15)(18,24)(21,29)(22,30)(23,25)(26,31)(28,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ef,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,c^b=ci,g^d=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300454);