Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×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 | |
1422 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)(5,6,8,9)(7,10,11,12)(21,22)(23,31)(24,28)(25,30)(26,29)(27,33)(32,36)(34,35);(6,9)(10,12)(13,14,16,17)(15,18,19,20)(21,23,25,33)(22,27,30,31)(24,34,26,32)(28,36,29,35);(2,4)(7,11)(10,12)(14,17)(18,20)(21,24,25,26)(22,28,30,29)(23,32,33,34)(27,35,31,36);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(22,29)(24,26)(27,35)(28,30)(31,36)(32,34)Polycyclic presentation:
2^9|a^2=ei,b^2=g,c^2=i,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377490);