Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | |
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)(3,8)(4,6)(5,7)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26)(27,32)(28,38)(29,34)(30,45)(31,35)(33,40)(36,48)(37,46)(39,54)(41,55)(42,49)(43,56)(44,51)(47,53)(50,52);(3,4)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,31,30)(26,32,35,45)(28,36,41,40)(29,42,44,37)(33,38,48,55)(34,49,51,46)(39,53,43,52)(47,56,50,54);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,30,40,31,41,27,36)(26,33,32,38,35,48,45,55)(29,43,42,52,44,39,37,53)(34,50,46,56,51,47,49,54);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,31,44)(26,34,35,51)(27,37,30,42)(28,39,41,43)(32,46,45,49)(33,47,48,50)(36,52,40,53)(38,54,55,56);(1,4,5,3)(2,6,7,8)(11,21)(15,18)(19,20)(23,24)(29,44)(34,51)(37,42)(39,43)(46,49)(47,50)(52,53)(54,56)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=fi,e^2=h,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^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6704560);