Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
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 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
990 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)(8,10)(9,11)(21,22,25,29,24,28,33,27)(23,30,32,36,31,26,34,35);(1,3,4,2)(5,7,9,8)(6,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,31)(25,32)(27,36)(28,30)(29,35)(33,34);(5,8,9,7)(6,10,11,12)(14,17)(18,20)(22,27)(23,31)(25,33)(26,35)(28,29)(30,36);(6,11)(10,12)(13,15,16,19)(14,18,17,20)(23,31)(26,30)(32,34)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=ei,d^2=gi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278692);