Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2921 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(9,10,13,17)(11,18,21,15)(12,22,14,16)(19,20,24,23);(1,2,4,3)(6,7)(9,10)(11,19)(12,16)(13,17)(14,22)(15,20)(18,23)(21,24);(5,6,8,7)(9,10)(11,15)(12,22)(13,17)(14,16)(18,21)(19,20)(23,24);(5,7,8,6)(9,11,13,21)(10,15,17,18)(12,20,14,23)(16,24,22,19);(1,3,4,2)(9,12,13,14)(10,16,17,22)(11,20,21,23)(15,24,18,19)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=g,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,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470311);