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)(23,26)(24,27)(25,28)(29,32)(30,35)(31,33)(34,36);(1,2,3,4)(6,9)(7,11)(13,15,16,19)(14,17,18,20)(21,23)(22,26)(24,31)(25,30)(27,33)(28,35)(29,34)(32,36);(2,4)(5,7)(6,10)(8,11)(9,12)(22,27)(24,32)(26,33)(28,29)(31,36)(34,35);(7,11)(10,12)(14,18)(17,20)(22,28)(23,30)(27,29)(31,36)Polycyclic presentation:
2^9|a^2=e,b^2=eghi,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, 378687);