Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 |
| prime radical | #59654 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2647 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)(9,10,14,17)(11,18,20,23)(12,16,13,22)(15,21,24,19);(2,4)(6,7)(9,11,12,21,14,20,13,19)(10,15,22,23,17,24,16,18);(5,6,8,7);(1,2,3,4)(5,7,8,6)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,24,18);(9,11,12,21,14,20,13,19)(10,15,22,23,17,24,16,18)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=h,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=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521266);