Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632088 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,29,34)(27,36,38,31)(28,40,41,33)(30,32,43,35)(37,45,50,49)(39,46,52,44)(42,48,53,47)(51,54,56,55);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27)(26,31)(28,37)(29,38)(30,39)(32,44)(33,45)(34,36)(35,46)(40,49)(41,50)(42,51)(43,52)(47,54)(48,55)(53,56);(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(26,32)(30,43)(31,44)(33,47)(34,35)(36,46)(39,52)(40,48)(42,53)(45,54)(49,55)(51,56);(2,5)(6,7)(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,28,29,41)(26,33,34,40)(27,37,38,50)(30,42,43,53)(31,45,36,49)(32,47,35,48)(39,51,52,56)(44,54,46,55);(1,3,4,8)(2,6,5,7)(10,18)(15,24)(16,17)(22,23)(26,34)(27,38)(28,41)(32,35)(39,52)(42,53)(45,49)(54,55)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=h,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6559795);