Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 | #34743 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10)(11,12)(13,22)(14,16)(15,17)(18,24)(19,20)(21,23);(5,6)(7,8)(9,11,14,19)(10,15,16,21)(12,18,20,22)(13,23,24,17);(2,4)(6,8)(10,16)(11,19)(17,23)(18,22);(2,4)(9,12,13,21,14,20,24,15)(10,17,18,19,16,23,22,11);(10,16)(12,20)(15,21)(18,22)Polycyclic presentation:
2^9|b^2=i,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,c^b=ch,d^b=df,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409337);