Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2833 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,8,7)(10,15)(11,17)(14,19)(16,23)(18,20)(21,24);(1,2)(3,4)(6,7)(9,10,13,15)(11,18,21,16)(12,19,22,14)(17,23,24,20);(2,4)(5,6,8,7)(9,11,19,24,13,21,14,17)(10,16,22,23,15,18,12,20);(2,4)(5,7,8,6)(9,12,13,22)(10,14,15,19)(11,20,21,23)(16,17,18,24);(10,15)(12,22)(16,18)(20,23)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351456);