Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3636 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)(9,10,13,17,14,15,12,22)(11,18,20,16,21,19,23,24);(5,6)(7,8)(9,10)(11,19)(12,17)(13,22)(14,15)(16,20)(18,21)(23,24);(2,4)(6,8)(10,15)(16,24)(17,22)(18,19);(2,4)(9,11)(10,16)(12,20)(13,23)(14,21)(15,24)(17,18)(19,22);(9,12,14,13)(10,17,15,22)(11,20,21,23)(16,18,24,19)Polycyclic presentation:
2^9|a^2=fgi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,f^b=fi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410540);