Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,31,35)(27,32,30,44)(28,38,40,52)(29,34,43,49)(33,36,47,41)(37,45,42,48)(39,51,54,55)(46,50,56,53);(1,2)(3,7)(4,6)(5,8)(10,14)(12,19)(15,20)(18,23)(25,27,31,30)(26,32,35,44)(28,36,40,41)(29,42,43,37)(33,38,47,52)(34,48,49,45)(39,53,54,50)(46,55,56,51);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,27,36,31,40,30,41)(26,33,44,52,35,47,32,38)(29,39,37,50,43,54,42,53)(34,46,48,55,49,56,45,51);(3,8)(5,7)(10,14)(11,17)(16,21)(18,23)(25,29)(26,34)(27,37)(28,39)(30,42)(31,43)(32,45)(33,46)(35,49)(36,50)(38,51)(40,54)(41,53)(44,48)(47,56)(52,55);(2,6)(5,7)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(28,40)(29,43)(33,47)(34,49)(36,41)(37,42)(38,52)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6819361);