Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1082 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)(5,6,9,8)(7,11,12,10)(13,14)(15,17)(16,18)(19,20)(21,22,25,30,26,31,27,32)(23,33,35,29,24,28,34,36);(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,17,18,20)(21,23,25,35,26,24,27,34)(22,28,30,36,31,33,32,29);(5,8,9,6)(7,11,12,10)(14,18)(17,20)(21,24,26,23)(22,29,31,36)(25,35,27,34)(28,30,33,32);(2,4)(7,12)(10,11)(21,25,26,27)(22,30,31,32)(23,34,24,35)(28,29,33,36)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340116);