Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7606 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,14,10,13,8,12)(7,15,16,20,17,11,18,19)(21,22)(23,30)(24,29)(25,27)(26,33)(28,31)(32,36)(34,35);(1,2,3,4)(22,27)(28,35)(29,33)(30,36);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)(21,23,25,32)(22,28,27,35)(24,31,26,34)(29,36,33,30);(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,24,25,26)(22,29,27,33)(23,31,32,34)(28,36,35,30)Polycyclic presentation:
2^9|a^2=g,b^2=ei,c^2=i,d^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,f^a=fh,f^c=fi,g^c=ghi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95810);