Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| 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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1279 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,8,4,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,34)(27,35,36,50)(28,32,40,47)(29,43,37,33)(31,44,45,39)(38,51,41,54)(42,56,52,49)(46,55,48,53);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(27,36)(28,40)(31,45)(32,47)(35,50)(39,44)(42,52)(49,56);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,37,44,30,36,29,39)(26,31,43,50,34,45,33,35)(28,41,52,53,40,38,42,55)(32,48,56,51,47,46,49,54);(12,20)(15,24)(17,19)(21,23)(25,28)(26,32)(27,38)(29,42)(30,40)(31,46)(33,49)(34,47)(35,51)(36,41)(37,52)(39,53)(43,56)(44,55)(45,48)(50,54);(1,4,5,3)(2,6,7,8)(11,18)(14,22)(17,19)(21,23)(28,40)(32,47)(38,41)(42,52)(46,48)(49,56)(51,54)(53,55)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6419798);