Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 | #2018 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1589 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18)(19,30,35,43)(20,36,32,46)(21,27)(22,41)(23,28)(24,29)(25,34,44,33)(26,38,47,39)(31,45)(37,42)(40,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19)(18,25)(20,37)(21,38)(22,34)(23,33)(24,35)(26,45)(27,46)(28,43)(29,44)(30,41)(31,32)(36,48)(39,40)(42,47);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,26)(19,31)(21,33)(22,38)(23,39)(24,32)(25,42)(27,30)(28,46)(29,47)(34,40)(35,37)(36,41)(43,48)(44,45);(3,10)(6,9)(11,16)(14,15)(17,21,22,37,24,40,23,31)(18,27,41,42,29,48,28,45)(19,32,33,39,35,20,34,38)(25,26,43,36,44,47,30,46);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(17,22,24,23)(18,28,29,41)(19,33,35,34)(20,38,32,39)(21,37,40,31)(25,30,44,43)(26,46,47,36)(27,45,48,42)Polycyclic presentation:
2^9|a^2=fgi,b^2=hi,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7090860);