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)(27,32)(28,38)(29,34)(30,45)(31,35)(33,40)(36,48)(37,46)(39,53)(41,54)(42,55)(43,50)(44,56)(47,52)(49,51);(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)(28,36,41,40)(29,37,43,44)(33,38,48,54)(34,46,50,56)(39,51,42,52)(47,53,49,55);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,27,36,31,41,30,40)(26,33,45,54,35,48,32,38)(29,42,37,52,43,39,44,51)(34,49,56,53,50,47,46,55);(1,3,5,4)(2,7,6,8)(12,20)(15,24)(17,19)(21,23)(25,29)(26,34)(27,37)(28,39)(30,44)(31,43)(32,46)(33,47)(35,50)(36,51)(38,53)(40,52)(41,42)(45,56)(48,49)(54,55);(1,4,5,3)(2,7,6,8)(10,16)(11,18)(15,24)(17,19)(29,43)(34,50)(37,44)(39,42)(46,56)(47,49)(51,52)(53,55)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gi,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=eh,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6957283);