Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| 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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10)(11,18)(12,16)(13,23)(14,17)(15,21)(19,20)(22,24);(5,6)(7,8)(9,11,12,20,14,22,13,21)(10,15,23,24,17,19,16,18);(2,4)(6,8)(9,10)(11,19)(12,23)(13,16)(14,17)(15,22)(18,20)(21,24);(5,6)(7,8)(9,12,14,13)(10,16,17,23)(11,20,22,21)(15,18,19,24);(6,8)(9,12,14,13)(10,16,17,23)(11,20,22,21)(15,18,19,24)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dh,e^c=ei,e^d=ehi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481258);