Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1682 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)(23,28)(24,36)(25,30)(26,40)(27,31)(29,38)(32,45)(33,42)(34,43)(35,49)(37,44)(39,50)(41,48)(46,52)(47,51);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23)(22,28)(24,32)(25,34)(26,35)(27,33)(29,41)(30,43)(31,42)(36,45)(37,47)(38,48)(39,46)(40,49)(44,51)(50,52);(2,4)(21,24,25,37,27,39,26,38)(22,29,40,50,31,44,30,36)(23,32,34,47,33,46,35,48)(28,41,49,52,42,51,43,45);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(23,33)(28,42)(32,46)(34,35)(41,51)(43,49)(45,52)(47,48);(2,4)(7,15)(10,13)(14,20)(18,19)(21,25,27,26)(22,30,31,40)(23,34,33,35)(24,37,39,38)(28,43,42,49)(29,36,44,50)(32,47,46,48)(41,45,51,52)Polycyclic presentation:
2^9|a^2=f,c^2=gi,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=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591281);