Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | |
1382 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,35)(27,31,37,44)(28,33,39,48)(29,32,41,34)(36,43,49,55)(38,42,51,45)(40,46,52,47)(50,53,56,54);(2,5)(3,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,37)(26,31,35,44)(28,36,39,49)(29,38,41,51)(32,42,34,45)(33,43,48,55)(40,50,52,56)(46,53,47,54);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,32)(28,39)(29,41)(31,42)(33,47)(34,35)(36,49)(38,51)(43,54)(44,45)(46,48)(53,55);(1,3,4,7)(2,6,5,8)(12,20)(15,24)(17,19)(21,23)(25,28,30,39)(26,33,35,48)(27,36,37,49)(29,40,41,52)(31,43,44,55)(32,46,34,47)(38,50,51,56)(42,53,45,54);(10,16)(11,18)(15,24)(17,19)(27,37)(28,39)(31,44)(33,48)(38,51)(40,52)(42,45)(46,47)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,d^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,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7011429);