Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #392 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,13)(14,16)(17,18,21,26,22,25,33,40)(19,23,29,36,28,35,42,46)(20,24,32,38,31,39,44,48)(27,34,41,45,30,37,43,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,30)(21,29)(22,28)(24,37)(25,35)(26,36)(27,31)(32,43)(33,42)(34,39)(38,47)(40,46)(41,44)(45,48);(10,13)(14,15)(17,20)(18,24)(19,27)(21,32)(22,31)(23,34)(25,39)(26,38)(28,30)(29,41)(33,44)(35,37)(36,45)(40,48)(42,43)(46,47);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(18,25)(20,31)(23,35)(26,40)(27,30)(32,44)(36,46)(41,43);(1,3,4,7)(2,6,5,8)(19,28)(23,35)(27,30)(29,42)(34,37)(36,46)(41,43)(45,47)Polycyclic presentation:
2^9|a^2=h,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=df,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6316643);