Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| 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 | #61420 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1614 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)(17,18,22,29,24,30,23,28)(19,31,33,45,34,44,37,25)(20,35,38,26,39,47,42,48)(21,40,36,43,41,27,32,46);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19)(18,25)(20,36)(21,38)(22,37)(23,33)(24,34)(26,46)(27,47)(28,31)(29,44)(30,45)(32,39)(35,40)(41,42)(43,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,26)(19,32)(21,33)(22,38)(23,42)(24,39)(25,43)(27,31)(28,35)(29,47)(30,48)(34,36)(37,41)(40,44)(45,46);(3,10)(6,9)(11,16)(14,15)(17,21,23,32,24,41,22,36)(18,27,28,43,30,40,29,46)(19,20,37,42,34,39,33,38)(25,26,44,35,45,48,31,47);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(17,22,24,23)(18,28,30,29)(19,33,34,37)(20,38,39,42)(21,36,41,32)(25,44,45,31)(26,35,48,47)(27,43,40,46)Polycyclic presentation:
2^9|a^2=f,b^2=hi,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7088171);