Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1296 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)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,23)(21,31)(22,26)(24,28)(25,29)(30,32);(5,6)(7,8)(9,11,12,16)(10,13,14,15)(19,28)(20,30)(23,32)(24,27);(1,2,3,4)(6,8)(17,19,29,30,22,28,21,20)(18,23,31,24,26,32,25,27);(17,20,21,28,22,30,29,19)(18,24,25,23,26,27,31,32)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=fi,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72265171, #72265225, #72265295, … |
| maximal normal subgroup of | #72265171, #72265225, #72265295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 320458);