Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1401 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)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,28)(19,29,33,24)(20,26,37,25)(21,38,23,27)(30,40,35,43)(31,47,48,41)(32,42,34,46)(36,44,39,45);(2,6)(7,8)(11,16)(13,15)(17,19,22,33)(18,24,28,29)(20,35,37,30)(21,34,23,32)(25,43,26,40)(27,46,38,42)(31,39,48,36)(41,45,47,44);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,30)(21,39)(22,37)(23,36)(24,40)(26,28)(27,45)(29,43)(31,34)(32,48)(33,35)(38,44)(41,46)(42,47);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,26)(19,31)(21,36)(22,37)(23,39)(24,41)(25,28)(27,45)(29,47)(30,34)(32,35)(33,48)(38,44)(40,46)(42,43);(1,4,5,3)(2,7,6,8)(17,21,22,23)(18,27,28,38)(19,32,33,34)(20,36,37,39)(24,42,29,46)(25,44,26,45)(30,31,35,48)(40,41,43,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7442723);