Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
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 | |
1258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,27)(24,29)(25,28)(26,30)(31,35)(32,34)(33,36);(1,3,4,2)(5,7,9,8)(6,10,12,11)(13,15,17,16)(14,18,20,19)(21,23,26,33)(22,27,30,36)(24,32,25,31)(28,35,29,34);(5,7,9,8)(6,11,12,10)(13,16,17,15)(14,19,20,18)(22,28)(23,31)(24,25)(29,30)(32,33)(34,35);(5,8,9,7)(6,11,12,10)(14,20)(18,19)(21,24,26,25)(22,29,30,28)(23,32,33,31)(27,34,36,35)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=e,d^2=efg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280088);