Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,30,35)(27,36,39,49)(28,33,40,47)(29,32,43,34)(31,42,45,38)(37,50,51,56)(41,46,53,48)(44,54,55,52);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,30,39)(26,31,35,45)(28,37,40,51)(29,42,43,38)(32,36,34,49)(33,44,47,55)(41,54,53,52)(46,50,48,56);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,19)(22,23)(26,32)(29,43)(31,36)(33,46)(34,35)(38,42)(41,53)(44,50)(45,49)(47,48)(52,54)(55,56);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,28,30,40)(26,33,35,47)(27,37,39,51)(29,41,43,53)(31,44,45,55)(32,46,34,48)(36,50,49,56)(38,52,42,54);(10,16)(14,21)(15,24)(22,23)(28,40)(33,47)(37,51)(41,53)(44,55)(46,48)(50,56)(52,54)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fgi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6392252);