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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1250 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)(3,7)(4,8)(5,6)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,30,34)(27,35,37,49)(28,39,40,32)(29,42,43,33)(31,44,46,38)(36,50,52,48)(41,47,55,54)(45,51,53,56);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,38,30,37,43,44)(26,31,33,35,34,46,42,49)(28,36,41,51,40,52,55,56)(32,45,47,48,39,53,54,50);(2,6)(7,8)(11,17)(14,20)(18,23)(21,24)(25,28)(26,32)(27,36)(29,41)(30,40)(31,45)(33,47)(34,39)(35,48)(37,52)(38,51)(42,54)(43,55)(44,56)(46,53)(49,50);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(27,37)(28,40)(31,46)(32,39)(35,49)(38,44)(41,55)(47,54);(1,4,5,3)(2,8,6,7)(12,19)(15,22)(18,23)(21,24)Polycyclic presentation:
2^9|a^2=hi,b^2=f,d^2=gi,e^2=gi,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, 6472830);