Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453966 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,8,9)(7,10,11,12)(13,14)(15,17)(16,18)(19,20)(21,22,25,29)(23,31,33,27)(24,30,26,28)(32,36,34,35);(1,2,3,4)(6,9)(7,11)(13,15,16,19)(14,17,18,20)(21,23,25,33)(22,27,29,31)(24,34,26,32)(28,35,30,36);(2,4)(5,7)(6,10)(8,11)(9,12)(22,28)(24,26)(27,35)(29,30)(31,36)(32,34);(7,11)(10,12)(14,18)(17,20)(21,24,25,26)(22,28,29,30)(23,32,33,34)(27,36,31,35)Polycyclic presentation:
2^9|a^2=e,b^2=eghi,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378688);