Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| 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 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1066 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)(7,11)(8,9)(10,12)(13,14)(15,19)(16,18)(17,20)(21,22,26,31,27,32,25,30)(23,28,33,29,24,34,35,36);(1,3,4,2)(6,9)(10,11)(13,15)(14,17)(16,20)(18,19)(21,23,27,24)(22,28,32,34)(25,35,26,33)(29,30,36,31);(15,20)(17,19)(21,24,27,23)(22,29,32,36)(25,35,26,33)(28,30,34,31);(5,7,8,12)(6,10,9,11)(14,18)(17,19)(21,25,27,26)(22,30,32,31)(23,33,24,35)(28,29,34,36)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=i,d^2=egi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,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, 278712);