Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1277 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)(27,32)(28,39)(29,34)(30,44)(31,35)(33,41)(36,50)(37,46)(38,48)(40,49)(42,53)(43,47)(45,51)(52,55)(54,56);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,38)(26,32,35,48)(28,36,42,52)(29,43,30,37)(33,45,49,56)(34,47,44,46)(39,50,53,55)(40,54,41,51);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,29,40,31,42,30,41)(26,33,44,53,35,49,34,39)(27,36,43,54,38,52,37,51)(32,45,46,55,48,56,47,50);(12,20)(15,24)(17,19)(21,23)(25,29,31,30)(26,34,35,44)(27,37,38,43)(28,40,42,41)(32,46,48,47)(33,39,49,53)(36,51,52,54)(45,55,56,50);(1,4,5,3)(2,6,7,8)(11,18)(14,22)(17,19)(21,23)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=gi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6419792);