Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,35)(28,39)(29,33)(30,34)(31,38)(32,36)(37,49)(40,48)(41,52)(42,46)(43,45)(44,50)(47,51)(53,56)(54,55);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,38)(26,31,34,35)(28,40,42,36)(29,43,44,37)(32,46,48,39)(33,49,50,45)(41,51,54,53)(47,52,56,55);(1,2,5,4)(3,6,8,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,38,40,30,42,27,36)(26,32,35,46,34,48,31,39)(29,41,43,53,44,54,37,51)(33,47,49,55,50,56,45,52);(1,2,5,4)(3,7,8,6)(12,20)(15,24)(17,19)(21,23)(25,29)(26,33)(27,37)(28,41)(30,44)(31,45)(32,47)(34,50)(35,49)(36,51)(38,43)(39,52)(40,53)(42,54)(46,55)(48,56);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(26,34)(28,42)(29,44)(31,35)(36,40)(37,43)(47,56)(52,55)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=fg,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6960034);