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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(6,10)(11,12)(13,14,17,21)(15,19,16,27)(18,22,28,23)(20,26,24,25);(1,3,4,2)(5,6,9,10)(7,11,8,12)(13,15)(14,19)(16,17)(18,26)(20,23)(21,27)(22,24)(25,28);(2,3)(5,7,9,8)(6,11,10,12)(13,14)(15,24)(16,20)(17,21)(18,23)(19,26)(22,28)(25,27);(5,8,9,7)(6,11,10,12)(13,16)(14,20)(15,17)(18,26)(19,23)(21,24)(22,27)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=e,d^2=e,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, 329772);