Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,19,20,18,15,16,22)(17,21,24,28,26,23,25,27);(1,2,3,4)(6,9)(10,11)(13,15,18,14)(16,22,19,20)(17,23,26,21)(24,28,25,27);(5,7,8,12)(6,10,9,11)(13,16,18,19)(14,20,15,22)(17,24,26,25)(21,27,23,28);(13,17,18,26)(14,21,15,23)(16,24,19,25)(20,27,22,28)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=f,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,e^b=ei,g^b=gi,h^b=hi,e^d=ei,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300510);