Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1271 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)(27,36)(28,41)(29,34)(30,35)(31,45)(32,38)(33,44)(37,51)(39,50)(40,47)(42,48)(43,56)(46,53)(49,54)(52,55);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,32)(28,37)(29,39)(30,38)(31,40)(33,46)(34,47)(35,36)(41,55)(42,53)(43,52)(44,54)(45,50)(48,49)(51,56);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,29,42,30,43,31,44)(26,33,45,56,35,48,34,41)(27,37,39,53,38,52,40,54)(32,46,50,51,36,49,47,55);(1,3,4,7)(2,6,5,8)(12,20)(15,24)(17,19)(21,23)(27,38)(32,36)(37,52)(39,40)(46,49)(47,50)(51,55)(53,54);(11,18)(14,22)(17,19)(21,23)(25,29,30,31)(26,34,35,45)(27,39,38,40)(28,42,43,44)(32,47,36,50)(33,41,48,56)(37,53,52,54)(46,55,49,51)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6418361);