Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1534 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)(11,14)(12,13)(15,16)(17,18,23,29)(19,30,32,25)(20,26,36,45)(21,38,24,27)(22,39,40,28)(31,46,47,41)(33,44,35,42)(34,48,37,43);(1,2)(3,6)(4,8)(5,7)(17,19,23,32)(18,25,29,30)(20,33,36,35)(21,22,24,40)(26,42,45,44)(27,39,38,28)(31,34,47,37)(41,48,46,43);(10,13)(14,16)(17,20,21,34,23,36,24,37)(18,26,38,48,29,45,27,43)(19,31,40,35,32,47,22,33)(25,41,39,44,30,46,28,42);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(17,21,23,24)(18,27,29,38)(19,22,32,40)(20,34,36,37)(25,39,30,28)(26,43,45,48)(31,33,47,35)(41,44,46,42);(1,4,5,3)(2,6,7,8)(17,22,21,19,23,40,24,32)(18,28,38,25,29,39,27,30)(20,35,34,47,36,33,37,31)(26,44,48,46,45,42,43,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=fg,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6735229);