Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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 | |
1054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,3)(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18,22,24,20,25,21,26)(19,27,30,32,28,23,29,31);(1,2,4,3)(5,7,8,6)(10,13)(11,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32);(1,2,4,3)(9,11,12,15)(10,14,13,16)(18,24)(21,22)(23,31)(25,26)(27,32)(29,30);(1,3,4,2)(10,13)(14,16)(19,28)(23,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=ei,d^2=egi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278802);