Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806827 |
| 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 | #30625 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1595 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)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18)(19,25)(20,21)(22,36)(23,27)(24,28)(26,39)(29,33)(30,45)(31,44)(32,41)(34,38)(35,43)(37,48)(40,47)(42,46);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,25)(20,33)(21,29)(22,37)(23,31)(24,32)(26,42)(27,44)(28,41)(30,38)(34,45)(35,47)(36,48)(39,46)(40,43);(17,20,23,18,24,27,21,28)(19,25,29,33,32,41,31,44)(22,34,38,26,39,43,35,36)(30,40,37,42,47,45,46,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,24,23)(18,20,28,27)(19,29,32,31)(22,35,39,38)(25,33,41,44)(26,34,36,43)(30,37,47,46)(40,42,45,48);(3,11)(6,9)(10,16)(14,15)(17,22,23,38,24,39,21,35)(18,26,27,43,28,36,20,34)(19,30,29,37,32,47,31,46)(25,40,33,42,41,45,44,48)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=fi,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715975);