Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,26)(21,27)(22,24)(23,28)(29,40)(30,39)(31,38)(32,36)(33,42)(34,37)(35,41)(43,48)(44,47)(45,46);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,24)(19,29)(20,33)(21,23)(25,38)(26,41)(31,32)(34,35)(36,40)(37,42)(39,46)(43,44);(10,14)(13,16)(17,19,22,32)(18,25,24,36)(20,30,34,45)(21,29,23,31)(26,39,37,46)(27,40,28,38)(33,43,35,44)(41,47,42,48);(1,4,5,3)(2,6,7,8)(17,20,22,34)(18,26,24,37)(19,30,32,45)(21,33,23,35)(25,39,36,46)(27,42,28,41)(29,43,31,44)(38,47,40,48);(17,21,22,23)(18,27,24,28)(19,31,32,29)(20,33,34,35)(25,38,36,40)(26,42,37,41)(30,44,45,43)(39,47,46,48)Polycyclic presentation:
2^9|a^2=fi,b^2=fgi,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348679);