Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
989 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)(5,6,9,8)(7,11,12,10)(13,14)(15,23)(16,20)(17,21)(18,24)(19,26)(22,27)(25,28);(5,7,9,12)(6,10,8,11)(13,14)(15,19)(16,20)(17,22)(18,25)(21,27)(23,26)(24,28);(1,2,3,4)(13,15,16,24)(14,18,20,23)(17,26,27,25)(19,21,28,22);(5,8,9,6)(7,10,12,11)(13,15,16,24)(14,19,20,28)(17,25,27,26)(18,21,23,22)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=f,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331205);