Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1387 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,29,31)(27,35,37,50)(28,39,41,33)(30,44,43,34)(32,46,45,38)(36,52,40,51)(42,49,53,55)(47,54,48,56);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(26,31)(32,45)(33,39)(34,44)(35,50)(47,48)(49,55)(51,52);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,27,30,38,29,37,43,46)(26,32,34,35,31,45,44,50)(28,40,42,56,41,36,53,54)(33,48,49,52,39,47,55,51);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,28)(26,33)(27,36)(29,41)(30,42)(31,39)(32,47)(34,49)(35,51)(37,40)(38,54)(43,53)(44,55)(45,48)(46,56)(50,52);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(28,41)(33,39)(36,40)(42,53)(47,48)(49,55)(51,52)(54,56)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6820190);