Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021363 |
| 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 | |
1648 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)(17,18,23,40,24,28,22,29)(19,30,39,25,32,43,33,44)(20,34,37,26,36,46,21,27)(31,41,48,42,35,45,38,47);(3,9)(4,12)(6,13)(7,11)(17,19)(18,25)(20,35)(21,38)(22,39)(23,33)(24,32)(26,45)(27,41)(28,44)(29,43)(30,40)(31,36)(34,47)(37,48)(42,46);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,20,22,21,24,36,23,37)(18,26,40,46,28,27,29,34)(19,31,39,48,32,35,33,38)(25,41,30,47,44,45,43,42);(3,9)(6,13)(10,15)(14,16)(17,21,22,36,24,37,23,20)(18,27,29,46,28,26,40,34)(19,31,33,38,32,35,39,48)(25,42,30,41,44,47,43,45);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,28)(19,32)(20,36)(21,37)(29,40)(33,39)(41,45)(42,47)Polycyclic presentation:
2^9|a^2=fh,c^2=gh,d^2=f,e^2=hi,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^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104033);