Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,7)(4,6)(5,8)(25,26,30,33)(27,31,36,45)(28,38,39,48)(29,41,43,34)(32,42,47,40)(35,49,51,56)(37,53,52,46)(44,54,55,50);(1,3,5,4)(2,6,8,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,31)(28,35)(29,37)(30,36)(32,44)(33,45)(34,46)(38,49)(39,51)(40,50)(41,53)(42,54)(43,52)(47,55)(48,56);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,28,30,39)(26,32,33,47)(27,35,36,51)(29,42,43,40)(31,44,45,55)(34,48,41,38)(37,54,52,50)(46,56,53,49);(1,3,5,4)(2,7,8,6)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(27,36)(28,39)(31,45)(32,47)(37,52)(38,48)(40,42)(46,53);(1,4,5,3)(2,6,8,7)(10,16)(14,17)(15,24)(22,23)(26,33)(31,45)(32,47)(34,41)(38,48)(44,55)(46,53)(49,56)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6437720);