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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1377 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)(27,35)(28,39)(29,31)(30,43)(32,38)(33,40)(34,42)(36,48)(37,51)(41,46)(44,50)(45,56)(47,49)(52,53)(54,55);(3,4)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(26,31)(27,36)(28,40)(34,43)(38,50)(41,55)(45,53)(47,51);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,30,38,29,36,42,50)(26,32,34,35,31,44,43,48)(28,37,41,52,40,49,55,56)(33,45,46,47,39,53,54,51);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,28,29,40)(26,33,31,39)(27,37,36,49)(30,41,42,55)(32,45,44,53)(34,46,43,54)(35,47,48,51)(38,52,50,56);(1,4,5,3)(2,6,7,8)(11,21)(15,18)(19,20)(23,24)(28,40)(33,39)(37,49)(41,55)(45,53)(46,54)(47,51)(52,56)Polycyclic presentation:
2^9|a^2=fi,c^2=fgi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,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, 6704523);