Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667748 |
| 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 | #3005 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1629 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)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,20)(21,25)(22,26)(23,28)(24,34)(27,37)(29,32)(30,33)(31,35)(36,42)(38,43)(39,44)(40,45)(41,47)(46,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,31)(18,20,28,35)(21,29,37,24)(22,38,40,41)(25,32,27,34)(26,43,45,47)(30,36,46,39)(33,42,48,44);(17,20)(18,24)(19,25)(21,32)(22,33)(23,35)(26,41)(27,31)(28,29)(30,42)(34,37)(36,43)(38,45)(39,47)(40,48)(44,46);(2,7)(6,9)(8,15)(14,16)(17,21,23,37)(18,25,28,27)(19,29,31,24)(20,32,35,34)(22,36,40,39)(26,42,45,44)(30,38,46,41)(33,43,48,47);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22)(18,26)(19,30)(20,33)(21,36)(23,40)(24,41)(25,42)(27,44)(28,45)(29,38)(31,46)(32,43)(34,47)(35,48)(37,39)Polycyclic presentation:
2^9|b^2=fi,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,g^c=gi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580786);