Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 | #36019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1691 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,27)(19,29,32,24)(20,25,23,28)(21,26,35,41)(30,43,45,37)(31,42,36,38)(33,39,34,40)(44,48,46,47);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19,22,32)(18,24,27,29)(20,31,23,36)(21,30,35,45)(25,38,28,42)(26,37,41,43)(33,44,34,46)(39,47,40,48);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,20)(18,25)(22,23)(27,28)(30,44)(31,36)(33,34)(37,47)(38,42)(39,40)(43,48)(45,46);(9,12,13,11)(10,16,15,14)(17,21,22,35)(18,26,27,41)(19,30,32,45)(20,33,23,34)(24,37,29,43)(25,39,28,40)(31,44,36,46)(38,47,42,48);(1,4,5,3)(2,6,8,7)(19,31)(20,23)(21,34)(24,38)(25,28)(26,40)(29,42)(30,45)(32,36)(33,35)(37,43)(39,41)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fi,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6497883);