Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1241 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,33,30,36,29,35)(27,32,44,46,39,49,38,48)(28,40,45,52,42,34,43,50)(37,51,54,56,41,47,53,55);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,30,39)(26,32,36,49)(28,41,42,37)(29,38,31,44)(33,46,35,48)(34,51,40,47)(43,54,45,53)(50,56,52,55);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(29,31)(32,46)(34,50)(35,36)(38,44)(40,52)(43,45)(47,55)(48,49)(51,56)(53,54);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,28)(26,34)(27,37)(29,43)(30,42)(31,45)(32,47)(33,50)(35,52)(36,40)(38,53)(39,41)(44,54)(46,55)(48,56)(49,51);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(28,42)(34,40)(37,41)(43,45)(47,51)(50,52)(53,54)(55,56)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=f,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6775997);