Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1362 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)(3,7)(4,6)(5,8)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26,31,35)(27,32,37,46)(28,40,42,48)(29,34,30,43)(33,44,49,41)(36,50,51,56)(38,47,39,53)(45,54,55,52);(1,3,4,8)(2,5,6,7)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27)(26,32)(28,36)(29,38)(30,39)(31,37)(33,45)(34,47)(35,46)(40,50)(41,52)(42,51)(43,53)(44,54)(48,56)(49,55);(3,8)(5,7)(9,10)(11,15)(12,21)(13,16)(14,17)(18,20)(19,24)(22,23)(25,28,31,42)(26,33,35,49)(27,36,37,51)(29,41,30,44)(32,45,46,55)(34,48,43,40)(38,52,39,54)(47,56,53,50);(2,6)(5,7)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(27,37)(32,46)(36,51)(38,39)(45,55)(47,53)(50,56)(52,54);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,29,31,30)(26,34,35,43)(27,38,37,39)(28,41,42,44)(32,47,46,53)(33,48,49,40)(36,52,51,54)(45,56,55,50)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=fh,e^2=fhi,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=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6893702);