Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 | |
1357 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)(30,44)(31,35)(33,40)(36,47)(37,45)(39,50)(41,51)(42,48)(43,54)(46,53)(49,55)(52,56);(3,4)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,31,30)(26,32,35,44)(28,36,41,40)(29,37,42,43)(33,38,47,51)(34,45,48,54)(39,49,52,53)(46,50,55,56);(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)(26,33,32,38,35,47,44,51)(29,39,43,53,42,52,37,49)(34,46,45,50,48,55,54,56);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29)(26,34)(27,37)(28,39)(30,43)(31,42)(32,45)(33,46)(35,48)(36,49)(38,50)(40,53)(41,52)(44,54)(47,55)(51,56);(1,4,5,3)(2,6,7,8)(11,21)(15,18)(19,20)(23,24)(29,42)(34,48)(37,43)(39,52)(45,54)(46,55)(49,53)(50,56)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6704438);