Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806827 |
| 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 | #525 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1531 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)(19,25)(20,23)(21,26)(22,37)(24,28)(27,40)(29,33)(30,41)(31,45)(32,34)(35,36)(38,48)(39,43)(42,47)(44,46);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,24,33)(18,25,28,29)(20,34,26,41)(21,30,23,32)(22,38,40,46)(27,44,37,48)(31,36,47,39)(35,42,43,45);(17,20,23,28,24,26,21,18)(19,29,30,34,33,25,32,41)(22,35,39,37,40,43,36,27)(31,45,38,48,47,42,46,44);(2,7)(6,9)(8,15)(14,16)(17,21,24,23)(18,26,28,20)(19,30,33,32)(22,36,40,39)(25,41,29,34)(27,43,37,35)(31,38,47,46)(42,44,45,48);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22,23,39,24,40,21,36)(18,27,20,35,28,37,26,43)(19,31,30,38,33,47,32,46)(25,42,41,44,29,45,34,48)Polycyclic presentation:
2^9|b^2=fi,c^2=h,d^2=i,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716871);