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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1729 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)(3,11)(4,8)(5,10)(6,9)(7,13)(12,16)(14,15)(17,18)(19,29)(20,27)(21,26)(22,25)(23,28)(24,32)(30,40)(31,45)(33,39)(34,48)(35,42)(36,41)(37,44)(38,47)(43,46);(1,3,6,13)(2,7,9,11)(4,12,5,14)(8,15,10,16)(17,19,21,32)(18,24,26,29)(20,34,35,46)(22,33,23,30)(25,40,28,39)(27,43,42,48)(31,36,47,37)(38,41,45,44);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(18,25)(19,30)(22,23)(26,28)(27,41)(31,46)(32,33)(34,47)(36,37)(39,40)(42,44)(43,48);(2,9)(7,11)(8,10)(15,16)(18,26)(24,29)(25,28)(27,42)(38,45)(39,40)(41,44)(43,48);(1,5,6,4)(2,10,9,8)(3,12,13,14)(7,15,11,16)(17,20,21,35)(18,27,26,42)(19,31,32,47)(22,36,23,37)(24,38,29,45)(25,41,28,44)(30,46,33,34)(39,48,40,43)Polycyclic presentation:
2^9|b^2=f,c^2=fi,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, 6582074);