Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1208 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)(9,10,14,17)(11,18,12,15)(13,21,22,16)(19,23,20,24)(25,26,30,34)(27,31,36,44)(28,33,39,48)(29,32,41,35)(37,43,49,55)(38,45,50,42)(40,46,52,47)(51,54,56,53);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,14)(13,20)(16,24)(17,18)(19,22)(21,23)(25,27)(26,31)(28,37)(29,38)(30,36)(32,45)(33,43)(34,44)(35,42)(39,49)(40,51)(41,50)(46,54)(47,53)(48,55)(52,56);(1,4,5,2)(3,7,8,6)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(26,32)(27,36)(28,39)(29,41)(31,42)(33,47)(34,35)(43,54)(44,45)(46,48)(51,56)(53,55);(3,8)(6,7)(11,12)(15,18)(19,20)(23,24)(25,28,30,39)(26,33,34,48)(27,37,36,49)(29,40,41,52)(31,43,44,55)(32,46,35,47)(38,51,50,56)(42,53,45,54);(9,13,14,22)(10,16,17,21)(11,19,12,20)(15,23,18,24)(26,34)(27,36)(28,39)(32,35)(38,50)(40,52)(43,55)(53,54)Polycyclic presentation:
2^9|a^2=fi,b^2=fhi,c^2=f,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7087508);