Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667760 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23956 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1649 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,23)(20,31)(21,29)(22,27)(24,26)(25,36)(28,33)(30,40)(32,35)(34,45)(37,38)(39,46)(41,43)(42,48)(44,47);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,18)(20,32,35,31)(21,29,24,26)(22,38,40,43)(25,33,28,36)(27,41,30,37)(34,42,39,44)(45,47,46,48);(17,20)(18,25)(19,28)(21,33)(22,39)(23,35)(24,36)(26,32)(27,44)(29,31)(30,42)(34,40)(37,48)(38,46)(41,47)(43,45);(2,7)(6,9)(8,15)(14,16)(17,21,23,24)(18,26,19,29)(20,33,35,36)(22,37,40,41)(25,32,28,31)(27,43,30,38)(34,47,39,48)(42,46,44,45);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22,23,40)(18,27,19,30)(20,34,35,39)(21,37,24,41)(25,42,28,44)(26,43,29,38)(31,45,32,46)(33,47,36,48)Polycyclic presentation:
2^9|b^2=f,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^c=ei,g^c=gi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580792);