Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| 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 | |
1374 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26)(27,31)(28,39)(29,33)(30,35)(32,41)(34,40)(36,49)(37,44)(38,45)(42,47)(43,52)(46,50)(48,53)(51,55)(54,56);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27)(26,31)(28,36)(29,38)(30,37)(32,43)(33,45)(34,46)(35,44)(39,49)(40,50)(41,52)(42,51)(47,55)(48,54)(53,56);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,40,53,30,42,29,41)(26,32,33,47,35,48,34,39)(27,36,50,56,37,51,38,52)(31,43,45,55,44,54,46,49);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(27,37)(31,44)(36,51)(38,50)(43,54)(45,46)(49,55)(52,56);(3,7)(5,8)(11,21)(15,18)(19,20)(23,24)(25,29,30,40)(26,33,35,34)(27,38,37,50)(28,41,42,53)(31,45,44,46)(32,47,48,39)(36,52,51,56)(43,55,54,49)Polycyclic presentation:
2^9|a^2=f,c^2=fghi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670883);