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 | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1116 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)(5,6,9,8)(7,11,12,10)(13,14)(15,19)(16,18)(17,20)(21,22,25,27)(23,28,31,34)(24,32,26,29)(30,36,33,35);(5,7)(6,10)(8,11)(9,12)(13,15,16,20)(14,17,18,19)(22,27)(23,30)(24,26)(28,35)(31,33)(34,36);(5,8,9,6)(7,11,12,10)(14,18)(17,19)(21,23)(22,28)(24,30)(25,31)(26,33)(27,34)(29,35)(32,36);(2,4)(7,12)(10,11)(15,20)(17,19)(21,24,25,26)(22,29,27,32)(23,30,31,33)(28,35,34,36)Polycyclic presentation:
2^9|a^2=efhi,b^2=f,c^2=efi,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357819);