Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1341 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)(9,10,13,12)(11,14,16,15)(17,18,24,29)(19,28,32,44)(20,26,35,43)(21,27,23,38)(22,37,31,25)(30,41,46,48)(33,40,36,42)(34,47,45,39);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,22,24,32,23,31)(18,25,38,44,29,37,27,28)(20,30,36,34,35,46,33,45)(26,39,40,48,43,47,42,41);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20)(18,26)(19,30)(21,36)(22,34)(23,33)(24,35)(25,39)(27,42)(28,41)(29,43)(31,45)(32,46)(37,47)(38,40)(44,48);(2,6)(5,7)(11,16)(14,15)(17,21,24,23)(18,27,29,38)(19,22,32,31)(20,33,35,36)(25,28,37,44)(26,40,43,42)(30,45,46,34)(39,48,47,41);(2,6)(5,7)(17,22,21,32,24,31,23,19)(18,28,38,25,29,44,27,37)(20,34,36,46,35,45,33,30)(26,41,40,39,43,48,42,47)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=h,d^2=i,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7514260);