Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1094 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)(7,8)(9,11)(10,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,30)(24,33)(25,27)(26,29)(28,32)(31,35)(34,36);(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,17,18,20)(22,27)(23,31)(24,26)(28,35)(30,36)(32,34);(5,8,9,10)(6,7,11,12)(14,18)(17,20)(21,23)(22,28)(24,31)(25,32)(26,34)(27,30)(29,35)(33,36);(6,11)(7,12)(14,18)(17,20)(21,24,25,26)(22,29,27,33)(23,31,32,34)(28,35,30,36)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=ei,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=defi,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, 291147);