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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(6,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,32)(27,37,38,49)(28,34,41,46)(29,36,31,35)(33,43,45,40)(39,51,52,56)(42,50,44,48)(47,53,55,54);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,32)(27,38)(28,41)(35,36)(40,43)(42,44)(47,55)(51,56);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,43,30,38,31,40)(26,33,35,49,32,45,36,37)(28,39,44,53,41,52,42,54)(34,47,50,56,46,55,48,51);(2,5)(6,8)(12,20)(15,24)(17,19)(21,23)(25,28)(26,34)(27,39)(29,44)(30,41)(31,42)(32,46)(33,47)(35,50)(36,48)(37,51)(38,52)(40,54)(43,53)(45,55)(49,56);(1,3,4,7)(2,6,5,8)(10,16)(11,18)(15,24)(17,19)(25,29,30,31)(26,35,32,36)(27,40,38,43)(28,42,41,44)(33,37,45,49)(34,48,46,50)(39,53,52,54)(47,56,55,51)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=gi,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^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6954403);