Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433049 |
| 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 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1665 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,37)(30,38)(31,40)(33,39)(35,42)(36,41)(43,45)(44,48)(46,47);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,32,23,33,22,31)(18,24,27,29,28,40,26,39)(20,30,36,46,35,44,37,45)(25,38,42,43,41,48,34,47);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20,22,37,23,35,21,36)(18,25,26,34,28,41,27,42)(19,30,31,45,33,44,32,46)(24,38,39,47,40,48,29,43);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(20,35)(25,41)(30,44)(34,42)(36,37)(38,48)(43,47)(45,46);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,35,37)(24,29,40,39)(25,34,41,42)(30,45,44,46)(38,43,48,47)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341483);