Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| 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 | #61415 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18,24,28,23,29,22,27)(19,20,32,25,33,36,31,35)(21,26,40,44,38,42,39,43)(30,34,46,41,45,47,37,48);(3,9)(4,12)(6,13)(7,11)(17,19)(18,25)(20,28)(21,37)(22,32)(23,33)(24,31)(26,34)(27,36)(29,35)(30,39)(38,46)(40,45)(41,43)(42,47)(44,48);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,20,22,35,23,36,24,25)(18,19,28,32,29,33,27,31)(21,34,39,48,38,47,40,41)(26,30,44,46,42,45,43,37);(3,9)(6,13)(10,15)(14,16)(17,21,22,39,23,38,24,40)(18,26,27,43,29,42,28,44)(19,30,31,37,33,45,32,46)(20,34,35,48,36,47,25,41);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(21,38)(26,42)(30,45)(34,47)(37,46)(39,40)(41,48)(43,44)Polycyclic presentation:
2^9|a^2=fhi,c^2=gh,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104022);