Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×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 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1482 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)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,18)(16,17)(19,20)(21,22,25,30,26,31,24,29)(23,27,32,28,34,36,33,35);(5,7,8,12)(6,9,10,11)(14,17)(15,19)(21,23)(22,27)(24,33)(25,32)(26,34)(28,30)(29,35)(31,36);(2,4)(13,15,16,19)(14,18,17,20)(21,23)(22,28)(24,32)(25,33)(26,34)(27,30)(29,36)(31,35);(6,10)(9,11)(14,17)(18,20)(21,24,26,25)(22,29,31,30)(23,32,34,33)(27,28,36,35)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=g,d^2=i,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, 270003);