Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,3)(5,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,24,20,25,21,26)(19,27,30,32,28,23,29,31);(1,2,4,3)(9,11,12,16)(10,13,14,15)(17,19,20,28)(18,23,25,27)(21,29,22,30)(24,31,26,32);(6,8)(11,16)(13,15)(18,24)(21,22)(23,31)(25,26)(27,32)(29,30);(1,3,4,2)(10,14)(13,15)(18,25)(19,28)(24,26)(29,30)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=egi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264657, #72264666, #72264751 |
| maximal normal subgroup of | #72264657, #72264666, #72264751 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270648);