Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1733 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22)(23,32)(24,29)(25,30)(26,41)(27,31)(28,35)(33,48)(34,43)(36,49)(37,44)(38,52)(39,46)(40,47)(42,50)(45,51);(1,2)(3,4)(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(24,37)(29,44)(33,38)(39,40)(42,45)(46,47)(48,52)(50,51);(2,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,23,26,35,27,36,25,34)(22,28,30,32,31,43,41,49)(24,38,39,51,37,33,40,50)(29,45,47,52,44,42,46,48);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,24,27,37)(22,29,31,44)(23,33,36,38)(25,40,26,39)(28,42,43,45)(30,47,41,46)(32,48,49,52)(34,51,35,50);(2,4)(7,17)(11,14)(15,16)(19,20)(21,25,27,26)(22,30,31,41)(23,34,36,35)(24,39,37,40)(28,32,43,49)(29,46,44,47)(33,50,38,51)(42,52,45,48)Polycyclic presentation:
2^9|c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591737);