Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 | |
2929 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(9,10,13,14)(11,15,19,18)(12,20,21,16)(17,23,24,22);(1,2,4,3)(6,7)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24);(5,6,8,7)(9,12,13,21)(10,16,14,20)(11,17,19,24)(15,22,18,23);(5,7,8,6)(10,14)(11,18)(15,19)(16,20)(17,23)(22,24);(1,3,4,2)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470264);