Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1430 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)(10,14)(11,12)(17,18,22,26)(19,29,31,24)(20,33,35,25)(21,38,32,27)(23,28,39,44)(30,40,47,45)(34,41,36,42)(37,48,46,43);(9,10)(11,16)(12,15)(13,14)(17,19,22,31)(18,24,26,29)(20,34,35,36)(21,39,32,23)(25,41,33,42)(27,44,38,28)(30,46,47,37)(40,48,45,43);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,15,14,16)(17,20,23,37,22,35,39,46)(18,25,28,43,26,33,44,48)(19,30,32,34,31,47,21,36)(24,40,38,41,29,45,27,42);(9,12,13,11)(10,15,14,16)(18,26)(20,35)(24,29)(27,38)(28,44)(30,47)(34,36)(37,46);(1,4,5,3)(2,6,7,8)(17,21,39,31,22,32,23,19)(18,27,44,29,26,38,28,24)(20,36,46,47,35,34,37,30)(25,42,48,45,33,41,43,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=fi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6744687);