Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
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 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,18)(17,20)(23,29)(24,31)(26,33)(28,35)(30,32)(34,36);(6,9)(10,11)(13,15,16,20)(14,17,18,19)(21,22)(23,26)(24,28)(25,27)(29,33)(30,34)(31,35)(32,36);(15,20)(17,19)(21,23,25,30)(22,26,27,34)(24,32,31,29)(28,36,35,33);(5,7,8,12)(6,10,9,11)(14,18)(17,19)(22,27)(26,34)(28,35)(33,36)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=eg,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271292);