Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | |
1431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,32)(28,38)(29,34)(30,44)(31,35)(33,39)(36,48)(37,45)(40,51)(41,46)(42,49)(43,54)(47,52)(50,56)(53,55);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,30)(26,32,35,44)(28,39,41,36)(29,37,42,43)(33,46,48,38)(34,45,49,54)(40,52,53,50)(47,55,56,51);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,39,31,41,27,36)(26,33,32,38,35,48,44,46)(29,40,43,52,42,53,37,50)(34,47,45,51,49,56,54,55);(3,8)(6,7)(12,20)(15,24)(17,19)(21,23)(25,29)(26,34)(27,37)(28,40)(30,43)(31,42)(32,45)(33,47)(35,49)(36,50)(38,51)(39,52)(41,53)(44,54)(46,55)(48,56);(10,16)(11,18)(15,24)(17,19)(26,35)(28,41)(29,42)(32,44)(36,39)(37,43)(47,56)(51,55)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=fg,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=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6948743);