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 | #2011 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1597 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18,24,29,23,30,22,28)(19,21,27,25,33,39,32,38)(20,34,40,43,36,44,37,26)(31,45,42,35,46,41,47,48);(1,3,6,2)(4,7,11,9)(5,8,13,10)(12,15,16,14)(17,19)(18,25)(20,31)(21,29)(22,27)(23,33)(24,32)(26,41)(28,39)(30,38)(34,35)(36,46)(37,42)(40,47)(43,45)(44,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20,22,37,23,36,24,40)(18,26,29,34,30,43,28,44)(19,31,27,42,33,46,32,47)(21,35,38,45,39,48,25,41);(4,11)(7,9)(12,16)(14,15)(17,21,22,38,23,39,24,25)(18,27,28,19,30,32,29,33)(20,35,37,45,36,48,40,41)(26,42,44,31,43,47,34,46);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(20,36)(26,43)(31,46)(34,44)(35,48)(37,40)(41,45)(42,47)Polycyclic presentation:
2^9|a^2=fhi,b^2=h,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^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7105842);