Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682638 |
| 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 | #60938 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1628 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)(19,24)(20,33)(21,26)(22,27)(23,29)(25,32)(28,36)(30,39)(31,40)(34,43)(35,42)(37,44)(38,45)(41,47)(46,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,24)(20,21)(22,34)(23,32)(25,29)(26,33)(27,43)(28,39)(30,36)(31,37)(35,47)(38,46)(40,44)(41,42)(45,48);(17,20,21,32,23,30,36,19)(18,25,28,33,29,24,26,39)(22,34,35,47,38,46,37,31)(27,41,44,43,45,40,42,48);(2,7)(6,9)(8,15)(14,16)(17,21,23,36)(18,26,29,28)(19,30,32,20)(22,35,38,37)(24,39,25,33)(27,42,45,44)(31,46,47,34)(40,48,41,43);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22)(18,27)(19,31)(20,34)(21,35)(23,38)(24,40)(25,41)(26,42)(28,44)(29,45)(30,46)(32,47)(33,43)(36,37)(39,48)Polycyclic presentation:
2^9|b^2=f,c^2=g,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,d^b=di,e^b=efh,h^b=hi,d^c=di,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, 6593324);