Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1182 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)(7,9)(8,10)(11,12)(13,14,16,18)(15,17,19,20)(21,22,26,28,25,30,24,29)(23,31,33,27,32,36,34,35);(5,7,8,11)(6,9,10,12)(15,19)(17,20)(21,23,25,32)(22,27,30,35)(24,33,26,34)(28,31,29,36);(6,10)(9,12)(13,15,16,19)(14,17,18,20)(23,32)(27,35)(31,36)(33,34);(2,4)(22,28)(23,33)(24,26)(29,30)(31,36)(32,34)Polycyclic presentation:
2^9|a^2=eg,b^2=fi,c^2=gi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351964);