Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| 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 | |
1323 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)(4,7)(5,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,31,35)(27,36,37,47)(28,33,41,50)(29,34,30,44)(32,42,45,39)(38,51,52,56)(40,48,43,49)(46,53,55,54);(1,3,6,2)(4,7,5,8)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21)(27,37)(32,45)(36,47)(38,52)(39,42)(46,55)(51,56)(53,54);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,30,39,31,37,29,42)(26,32,34,36,35,45,44,47)(28,38,40,54,41,52,43,53)(33,46,49,51,50,55,48,56);(1,4,6,5)(2,7,3,8)(12,20)(15,24)(17,19)(21,23)(25,28)(26,33)(27,38)(29,43)(30,40)(31,41)(32,46)(34,49)(35,50)(36,51)(37,52)(39,54)(42,53)(44,48)(45,55)(47,56);(1,5,6,4)(2,7,3,8)(11,18)(14,22)(17,19)(21,23)(25,29,31,30)(26,34,35,44)(27,39,37,42)(28,40,41,43)(32,47,45,36)(33,48,50,49)(38,53,52,54)(46,51,55,56)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=g,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7178905);