Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 | |
1724 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)(21,22,26,31)(23,32,36,42)(24,28,37,44)(25,29,33,30)(27,40,43,35)(34,47,48,52)(38,45,39,46)(41,49,51,50);(1,2)(3,4)(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(24,37)(28,44)(34,48)(38,39)(41,51)(45,46)(47,52)(49,50);(2,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,23,33,40,26,36,25,35)(22,27,29,42,31,43,30,32)(24,34,38,49,37,48,39,50)(28,41,46,52,44,51,45,47);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,24)(22,28)(23,34)(25,39)(26,37)(27,41)(29,46)(30,45)(31,44)(32,47)(33,38)(35,50)(36,48)(40,49)(42,52)(43,51);(2,4)(7,17)(11,14)(15,16)(19,20)(21,25,26,33)(22,29,31,30)(23,35,36,40)(24,38,37,39)(27,42,43,32)(28,45,44,46)(34,49,48,50)(41,47,51,52)Polycyclic presentation:
2^9|a^2=i,c^2=fgi,d^2=f,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,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591703);