Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 | C2×#3271 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,15,20)(11,22,23,17)(12,24,13,18)(14,21,16,19);(1,2,3,4)(5,6)(7,8)(9,11,15,23)(10,17,20,22)(12,14,13,16)(18,21,24,19);(6,8)(9,12)(10,17)(11,16)(13,15)(14,23)(18,21)(19,24)(20,22);(5,6)(7,8)(9,13,15,12)(10,18,20,24)(11,14,23,16)(17,19,22,21);(6,8)(9,14,15,16)(10,19,20,21)(11,12,23,13)(17,24,22,18)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481593);