Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1216 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)(25,26,30,32)(27,37,38,48)(28,34,42,46)(29,36,31,35)(33,40,45,41)(39,51,52,56)(43,50,44,49)(47,54,55,53);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,32)(27,38)(28,42)(35,36)(40,41)(43,44)(47,55)(51,56);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,40,30,38,31,41)(26,33,35,48,32,45,36,37)(28,39,44,54,42,52,43,53)(34,47,50,56,46,55,49,51);(1,2,5,4)(3,7,8,6)(12,20)(15,24)(17,19)(21,23)(25,28)(26,34)(27,39)(29,44)(30,42)(31,43)(32,46)(33,47)(35,50)(36,49)(37,51)(38,52)(40,54)(41,53)(45,55)(48,56);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(25,29,30,31)(26,35,32,36)(27,40,38,41)(28,43,42,44)(33,48,45,37)(34,49,46,50)(39,53,52,54)(47,51,55,56)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,c^2=gi,d^2=fi,e^2=f,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^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6958965);