Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | |
1147 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26,30,35)(27,32,38,46)(28,39,41,33)(29,34,43,49)(31,36,44,50)(37,51,53,45)(40,54,56,47)(42,55,52,48);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,38)(26,32,35,46)(28,40,41,56)(29,31,43,44)(33,47,39,54)(34,36,49,50)(37,52,53,42)(45,55,51,48);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,31,42,30,41,44,52)(26,33,36,48,35,39,50,55)(27,37,29,40,38,53,43,56)(32,45,34,47,46,51,49,54);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(28,41)(33,39)(37,53)(40,56)(42,52)(45,51)(47,54)(48,55);(1,3,4,7)(2,6,5,8)(11,21)(15,18)(19,20)(23,24)(25,29,31,38,30,43,44,27)(26,34,36,46,35,49,50,32)(28,40,42,53,41,56,52,37)(33,47,48,51,39,54,55,45)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=fh,d^2=f,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6741172);