Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(21,22,27,40,26,31,25,30)(23,32,34,28,36,43,35,44)(24,37,41,45,38,29,39,52)(33,47,49,51,48,46,50,42);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,26,36)(22,28,31,44)(24,33,38,48)(25,34,27,35)(29,42,37,51)(30,43,40,32)(39,49,41,50)(45,47,52,46);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24,26,38)(22,29,31,37)(23,33,36,48)(25,39,27,41)(28,42,44,51)(30,45,40,52)(32,46,43,47)(34,49,35,50);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(24,38)(29,37)(33,48)(39,41)(42,51)(45,52)(46,47)(49,50);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,26,27)(22,30,31,40)(23,34,36,35)(24,39,38,41)(28,43,44,32)(29,45,37,52)(33,49,48,50)(42,47,51,46)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,d^c=di,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451344);