Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1664 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,22,27)(19,23,29,38)(20,32,36,42)(21,25,34,26)(24,40,43,35)(28,45,33,48)(30,39,31,44)(37,46,41,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,33)(21,31)(22,29)(24,41)(25,44)(26,39)(27,38)(28,36)(30,34)(32,48)(35,46)(37,43)(40,47)(42,45);(2,6)(3,7)(10,13)(14,16)(17,20,34,40,22,36,21,35)(18,24,25,42,27,43,26,32)(19,28,30,46,29,33,31,47)(23,37,44,48,38,41,39,45);(9,11,12,15)(10,14,13,16)(19,29)(23,38)(28,33)(30,31)(37,41)(39,44)(45,48)(46,47);(3,7)(5,8)(17,21,22,34)(18,25,27,26)(19,30,29,31)(20,35,36,40)(23,39,38,44)(24,42,43,32)(28,46,33,47)(37,45,41,48)Polycyclic presentation:
2^9|a^2=i,c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670699);