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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1692 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)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22)(23,31)(24,35)(25,27)(26,40)(28,34)(29,36)(30,39)(32,46)(33,48)(37,45)(38,44)(41,47)(42,52)(43,50)(49,51);(1,2)(3,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(22,27)(23,32)(24,36)(30,40)(34,47)(38,49)(42,43)(45,48);(2,4)(21,23,26,34,25,32,39,47)(22,28,30,31,27,41,40,46)(24,37,38,52,36,33,49,50)(29,43,44,48,35,42,51,45);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24)(22,29)(23,33)(25,36)(26,38)(27,35)(28,42)(30,44)(31,45)(32,37)(34,50)(39,49)(40,51)(41,43)(46,48)(47,52);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(24,36)(29,35)(33,37)(38,49)(42,43)(44,51)(45,48)(50,52)Polycyclic presentation:
2^9|c^2=g,d^2=fi,e^2=fi,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, 6593761);