Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1954 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,18,22,26)(19,23,29,38)(20,24,32,41)(21,25,35,44)(27,36,45,47)(28,37,30,39)(31,40,46,48)(33,42,34,43);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,22,29)(18,23,26,38)(20,31,32,46)(21,28,35,30)(24,40,41,48)(25,37,44,39)(27,34,45,33)(36,43,47,42);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20)(18,24)(19,27)(21,34)(22,32)(23,36)(25,43)(26,41)(28,31)(29,45)(30,46)(33,35)(37,40)(38,47)(39,48)(42,44);(1,4,5,3)(2,8,6,7)(19,28)(21,35)(23,37)(25,44)(27,31)(29,30)(33,34)(36,40)(38,39)(42,43)(45,46)(47,48);(19,29)(20,32)(23,38)(24,41)(28,30)(33,34)(37,39)(42,43)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354239);