Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,31,35)(27,36,38,50)(28,33,41,47)(29,34,30,44)(32,39,46,40)(37,49,53,56)(42,48,43,54)(45,51,55,52);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,39,31,38,30,40)(26,32,44,50,35,46,34,36)(28,37,42,51,41,53,43,52)(33,45,54,56,47,55,48,49);(2,5)(6,8)(11,17)(14,20)(18,23)(21,24)(25,28)(26,33)(27,37)(29,42)(30,43)(31,41)(32,45)(34,48)(35,47)(36,49)(38,53)(39,51)(40,52)(44,54)(46,55)(50,56);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(27,38)(28,41)(32,46)(33,47)(36,50)(39,40)(42,43)(48,54);(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,44)(27,39,38,40)(28,42,41,43)(32,36,46,50)(33,48,47,54)(37,51,53,52)(45,49,55,56)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6472427);