Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518904 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,7)(9,10,15,19)(11,18,21,24)(12,17,14,23)(13,22,20,16);(2,4)(6,7)(9,11,14,20,15,21,12,13)(10,16,17,18,19,22,23,24);(5,6,8,7);(1,2,3,4)(5,7,8,6)(9,12,15,14)(10,17,19,23)(11,13,21,20)(16,18,22,24);(9,13,12,21,15,20,14,11)(10,18,23,16,19,24,17,22)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=hi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521222);