Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,25)(19,23,30,38)(20,24,31,41)(21,26,35,42)(27,36,45,48)(28,37,46,47)(29,39,33,43)(32,40,34,44);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,27)(21,33)(22,30)(24,36)(25,38)(26,43)(28,32)(29,35)(31,45)(34,46)(37,40)(39,42)(41,48)(44,47);(10,14)(11,16)(17,20)(18,24)(19,27)(21,32)(22,31)(23,36)(25,41)(26,40)(28,33)(29,46)(30,45)(34,35)(37,43)(38,48)(39,47)(42,44);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,20)(18,24)(19,28)(21,34)(22,31)(23,37)(25,41)(26,44)(27,33)(29,45)(30,46)(32,35)(36,43)(38,47)(39,48)(40,42);(2,7)(6,8)(10,14)(13,15)(18,25)(20,31)(23,38)(26,42)(27,45)(28,46)(32,34)(39,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7456203);