Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,29,30,24)(20,26,34,44)(21,28,23,25)(31,42,33,39)(32,38,36,40)(35,41,37,43)(45,48,46,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,24,27,29)(20,33,34,31)(21,36,23,32)(25,40,28,38)(26,42,44,39)(35,45,37,46)(41,47,43,48);(9,11,12,16)(10,13,14,15)(18,25)(19,30)(20,34)(21,23)(24,38)(26,43)(27,28)(29,40)(39,48)(41,44)(42,47)(45,46);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20,22,34)(18,26,27,44)(19,31,30,33)(21,37,23,35)(24,39,29,42)(25,41,28,43)(32,46,36,45)(38,47,40,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,23)(18,25,27,28)(19,32,30,36)(20,35,34,37)(24,38,29,40)(26,43,44,41)(31,45,33,46)(39,48,42,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6967831);