Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,6)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,31,35)(27,32,30,45)(28,38,41,47)(29,34,43,51)(33,36,49,39)(37,46,44,56)(40,54,42,55)(48,52,50,53);(1,2)(3,8)(4,7)(5,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,30)(26,32,35,45)(28,39,41,36)(29,37,43,44)(33,47,49,38)(34,46,51,56)(40,53,42,52)(48,55,50,54);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,39,31,41,27,36)(26,33,32,38,35,49,45,47)(29,42,44,52,43,40,37,53)(34,50,46,55,51,48,56,54);(1,3,5,4)(2,7,6,8)(12,20)(15,24)(17,19)(21,23)(25,29)(26,34)(27,37)(28,40)(30,44)(31,43)(32,46)(33,48)(35,51)(36,52)(38,54)(39,53)(41,42)(45,56)(47,55)(49,50);(1,4,5,3)(2,7,6,8)(10,16)(11,18)(15,24)(17,19)(26,35)(29,43)(32,45)(33,49)(37,44)(38,47)(40,42)(52,53)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6957151);