Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476082 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1236 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)(5,6,9,11)(7,12,8,10)(13,14,16,18)(15,17,19,20)(21,22)(23,28)(24,29)(25,27)(26,30)(31,36)(32,35)(33,34);(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)(22,27)(23,31)(24,26)(28,35)(32,33)(34,36);(5,8,9,7)(6,10,11,12)(14,18)(17,20)(21,23,25,33)(22,28,27,34)(24,31,26,32)(29,36,30,35);(2,4)(7,8)(10,12)(15,19)(17,20)(21,24,25,26)(22,29,27,30)(23,32,33,31)(28,35,34,36)Polycyclic presentation:
2^9|a^2=efi,b^2=ei,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383794);