Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 | |
1338 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)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,36)(28,33)(29,34)(30,43)(31,35)(32,37)(38,50)(39,45)(40,48)(41,49)(42,53)(44,47)(46,51)(52,55)(54,56);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,31,39)(26,32,35,47)(28,37,30,44)(29,38,41,52)(33,45,43,36)(34,46,49,56)(40,51,42,54)(48,55,53,50);(2,5)(6,8)(11,17)(12,19)(14,20)(15,22)(25,28,31,30)(26,33,35,43)(27,37,39,44)(29,40,41,42)(32,45,47,36)(34,48,49,53)(38,51,52,54)(46,55,56,50);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(25,29,31,41)(26,34,35,49)(27,38,39,52)(28,40,30,42)(32,46,47,56)(33,48,43,53)(36,50,45,55)(37,51,44,54);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(27,39)(29,41)(32,47)(34,49)(36,45)(37,44)(40,42)(48,53)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,f^b=fi,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436043);