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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,36)(28,41)(29,35)(30,32)(31,46)(33,40)(34,44)(37,54)(38,47)(39,52)(42,53)(43,51)(45,50)(48,56)(49,55);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(26,32)(33,47)(34,41)(35,46)(36,52)(48,49)(50,51)(53,54);(1,3)(2,5)(4,8)(6,7)(12,21)(15,20)(18,24)(22,23)(25,27,31,40,30,39,29,38)(26,33,35,36,32,47,46,52)(28,42,43,56,44,37,45,55)(34,49,51,54,41,48,50,53);(3,8)(5,7)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28,30,44)(26,34,32,41)(27,37,39,42)(29,45,31,43)(33,48,47,49)(35,51,46,50)(36,53,52,54)(38,56,40,55);(2,6)(5,7)(11,19)(14,17)(18,24)(22,23)(25,29,30,31)(26,35,32,46)(27,38,39,40)(28,43,44,45)(33,36,47,52)(34,50,41,51)(37,55,42,56)(48,54,49,53)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=g,d^2=f,e^2=i,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, 6817726);