Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | C2×#555 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1459 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,33)(21,36)(22,28)(23,29)(24,30)(26,35)(27,39)(31,46)(32,42)(34,43)(37,48)(38,45)(40,44)(41,47);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,32)(18,25,29,42)(20,24,35,22)(21,37,39,34)(26,28,33,30)(27,43,36,48)(31,38,47,40)(41,44,46,45);(1,4,5,3)(2,6,7,8)(10,14)(11,15)(17,20)(18,26)(19,22)(21,34)(23,35)(24,32)(25,30)(27,43)(28,42)(29,33)(31,40)(36,48)(37,39)(38,47)(41,44)(45,46);(10,14)(13,16)(17,21,23,39)(18,27,29,36)(19,31,32,47)(20,34,35,37)(22,40,24,38)(25,41,42,46)(26,43,33,48)(28,45,30,44);(2,7)(6,8)(17,22,23,24)(18,28,29,30)(19,20,32,35)(21,38,39,40)(25,33,42,26)(27,44,36,45)(31,37,47,34)(41,43,46,48)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7431438);