Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,12)(8,11)(9,10)(13,14)(15,19)(16,17)(18,20)(21,22,24,28,25,27,33,36)(23,29,31,35,30,34,32,26);(1,3,4,2)(6,10)(11,12)(14,17)(15,20)(21,23)(22,26)(24,32)(25,30)(27,35)(28,34)(29,36)(31,33);(5,7,9,8)(6,11,10,12)(13,15,16,20)(14,18,17,19)(22,27)(23,30)(28,36)(31,32);(5,8,9,7)(6,12,10,11)(14,17)(18,19)(22,28)(24,33)(26,34)(27,36)(29,35)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=fi,d^2=fg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=ei,h^b=hi,e^d=ei,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280717);