Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26)(27,32)(28,40)(29,34)(30,35)(31,36)(33,39)(37,46)(38,47)(41,51)(42,54)(43,48)(44,49)(45,50)(52,55)(53,56);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,38)(26,32,35,47)(28,41,39,31)(29,43,44,42)(33,36,40,51)(34,48,49,54)(37,52,53,45)(46,55,56,50);(1,3,4,8)(2,5,6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,31,27,30,39,41,38)(26,33,36,47,35,40,51,32)(29,42,45,37,44,43,52,53)(34,48,50,56,49,54,55,46);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(29,44)(34,49)(37,53)(42,43)(45,52)(46,56)(48,54)(50,55);(11,21)(15,18)(19,20)(23,24)(25,29,31,45,30,44,41,52)(26,34,36,50,35,49,51,55)(27,37,39,43,38,53,28,42)(32,46,33,48,47,56,40,54)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=fghi,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6725301);