Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1420 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)(17,18,21,26)(19,28,30,23)(20,25,32,39)(22,24,34,27)(29,41,42,36)(31,37,44,35)(33,38,45,40)(43,47,48,46);(1,3,5,4)(2,6,7,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,23)(20,29)(21,30)(22,31)(24,35)(25,36)(26,28)(27,37)(32,42)(33,43)(34,44)(38,46)(39,41)(40,47)(45,48);(9,11,13,12)(10,15,14,16)(18,24)(22,34)(23,35)(25,38)(26,27)(28,37)(31,44)(33,45)(36,46)(39,40)(41,47)(43,48);(1,4,5,3)(2,6,7,8)(9,12,13,11)(10,15,14,16)(17,20,21,32)(18,25,26,39)(19,29,30,42)(22,33,34,45)(23,36,28,41)(24,38,27,40)(31,43,44,48)(35,46,37,47);(19,30)(20,32)(23,28)(25,39)(31,44)(33,45)(35,37)(38,40)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=h,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342064);