Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417575 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1509 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)(15,21,22,24)(18,25,26,28);(2,3)(5,6,9,10)(7,11,8,12)(13,14,17,20)(15,18,22,26)(16,19,23,27)(21,25,24,28);(1,2,4,3)(5,7,9,8)(6,11,10,12)(16,23)(19,27)(21,24)(25,28);(1,3,4,2)(5,8,9,7)(6,11,10,12)(13,15)(14,18)(16,24)(17,22)(19,28)(20,26)(21,23)(25,27)Polycyclic presentation:
2^9|a^2=fg,b^2=egi,c^2=e,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348247);