Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,10,9)(7,12,8,11)(15,19)(17,20)(21,22,26,29)(23,32,24,35)(25,31,27,30)(28,34,36,33);(5,7,10,8)(6,11,9,12)(13,14,16,18)(15,20,19,17)(21,23,27,34,26,24,25,33)(22,28,31,35,29,36,30,32);(2,4)(5,8,10,7)(6,11,9,12)(13,15,16,19)(14,17,18,20)(22,29)(23,33)(24,34)(25,27)(28,35)(32,36);(5,9,10,6)(7,12,8,11)(21,24,25,34,26,23,27,33)(22,28,30,32,29,36,31,35)Polycyclic presentation:
2^9|a^2=efg,b^2=egh,c^2=eg,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 382048);