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 | |
1340 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)(30,44)(31,35)(32,37)(38,52)(39,46)(40,49)(41,55)(42,51)(43,56)(45,48)(47,53)(50,54);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,31,39)(26,32,35,48)(28,37,30,45)(29,41,42,38)(33,46,44,36)(34,50,51,47)(40,54,43,53)(49,52,56,55);(2,5)(6,8)(11,17)(12,19)(14,20)(15,22)(25,28,31,30)(26,33,35,44)(27,37,39,45)(29,40,42,43)(32,46,48,36)(34,49,51,56)(38,53,41,54)(47,55,50,52);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(25,29,31,42)(26,34,35,51)(27,38,39,41)(28,40,30,43)(32,47,48,50)(33,49,44,56)(36,52,46,55)(37,53,45,54);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(27,39)(29,42)(32,48)(34,51)(36,46)(37,45)(40,43)(49,56)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,d^b=di,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, 6436107);