Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1354 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,31,35)(27,32,38,48)(28,39,41,49)(29,34,30,44)(33,43,50,40)(36,51,53,56)(37,46,42,47)(45,52,55,54);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27)(26,32)(28,36)(29,42)(30,37)(31,38)(33,45)(34,47)(35,48)(39,51)(40,54)(41,53)(43,52)(44,46)(49,56)(50,55);(2,6)(3,7)(25,28,29,43,31,41,30,40)(26,33,44,49,35,50,34,39)(27,36,42,52,38,53,37,54)(32,45,46,56,48,55,47,51);(10,15)(14,17)(16,23)(22,24)(25,29,31,30)(26,34,35,44)(27,37,38,42)(28,40,41,43)(32,46,48,47)(33,49,50,39)(36,52,53,54)(45,51,55,56);(3,7)(5,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(28,41)(33,50)(36,53)(39,49)(40,43)(45,55)(51,56)(52,54)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=ghi,d^2=i,e^2=f,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^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6672278);