Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500450 |
| 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 | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(17,18,22,24)(19,28,31,42)(20,26,34,38)(21,27,23,36)(25,30,37,32)(29,43,45,48)(33,40,35,41)(39,46,47,44);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,24)(25,37)(26,38)(27,36)(28,42)(39,47)(40,41)(43,48);(9,11)(10,13)(12,16)(14,15)(17,19,23,32,22,31,21,30)(18,25,27,28,24,37,36,42)(20,29,33,44,34,45,35,46)(26,39,41,43,38,47,40,48);(11,16)(13,15)(17,20)(18,26)(19,29)(21,35)(22,34)(23,33)(24,38)(25,39)(27,41)(28,43)(30,46)(31,45)(32,44)(36,40)(37,47)(42,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,22,23)(18,27,24,36)(19,30,31,32)(20,33,34,35)(25,28,37,42)(26,40,38,41)(29,44,45,46)(39,48,47,43)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6415478);