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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,27,31)(23,32,36,49)(24,29,37,44)(25,30,26,41)(28,34,43,35)(33,48,38,52)(39,46,40,47)(42,51,45,50);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(24,37)(29,44)(33,38)(39,40)(42,45)(46,47)(48,52)(50,51);(2,4)(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);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(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)(7,15)(10,13)(14,20)(18,19)(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|a^2=fi,c^2=g,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, 6591337);