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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,6,4,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,29,34,30,35,43,31)(27,36,39,44,38,32,46,48)(28,40,41,45,42,33,49,50)(37,47,52,56,53,51,55,54);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,31)(28,41)(29,43)(32,44)(34,35)(36,48)(37,52)(39,46)(42,49)(45,50)(53,55)(54,56);(1,3,5,4)(2,7,8,6)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,27)(26,32)(28,37)(29,39)(30,38)(31,44)(33,47)(34,48)(35,36)(40,51)(41,52)(42,53)(43,46)(45,54)(49,55)(50,56);(1,3,5,4)(2,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,28,29,41,30,42,43,49)(26,33,34,50,35,40,31,45)(27,37,39,52,38,53,46,55)(32,47,48,56,36,51,44,54);(1,4,5,3)(2,7,8,6)(10,16)(11,18)(15,24)(17,19)(27,38)(28,42)(32,36)(33,40)(39,46)(41,49)(44,48)(45,50)Polycyclic presentation:
2^9|a^2=fghi,b^2=gi,c^2=ghi,d^2=fgi,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 7031166);