Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500740 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2815 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,14,18)(11,19,21,16)(12,22,23,20)(13,24,15,17);(1,3,4,8)(2,5,6,7)(9,11,14,21)(10,16,18,19)(12,17,23,24)(13,22,15,20);(9,12)(10,11)(13,19)(14,23)(15,16)(17,20)(18,21)(22,24);(2,6)(3,8)(9,11)(10,12)(13,20)(14,21)(15,22)(16,24)(17,19)(18,23);(9,13,14,15)(10,17,18,24)(11,20,21,22)(12,19,23,16)Polycyclic presentation:
2^9|a^2=f,b^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^b=ei,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213751, #72213818, #72228064, … |
| maximal normal subgroup of | #72213751, #72213818, #72228064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408342);