Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | #31021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,19,28,22,27,23,24)(20,31,32,25,34,38,29,41)(21,35,30,39,36,26,37,42)(33,46,45,48,47,44,43,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,30,36,37)(25,31,41,38)(26,39,35,42)(33,45,47,43)(40,44,48,46);(10,14)(15,16)(17,20,22,34)(18,25,27,41)(19,29,23,32)(21,33,36,47)(24,38,28,31)(26,40,35,48)(30,43,37,45)(39,46,42,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,22,36)(18,26,27,35)(19,30,23,37)(20,33,34,47)(24,39,28,42)(25,40,41,48)(29,43,32,45)(31,44,38,46);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(18,27)(21,36)(24,28)(25,41)(30,37)(31,38)(33,47)(43,45)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6775154);