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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18,22,37,23,27,21,26)(19,28,30,24,32,39,31,40)(20,25,35,47,36,43,34,42)(29,41,44,38,45,48,33,46);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19)(18,24)(20,33)(21,30)(22,31)(23,32)(25,41)(26,39)(27,40)(28,37)(29,34)(35,45)(36,44)(38,42)(43,48)(46,47);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,18)(19,24)(20,25)(21,26)(22,37)(23,27)(28,31)(29,38)(30,39)(32,40)(33,41)(34,42)(35,47)(36,43)(44,48)(45,46);(3,10)(6,9)(11,16)(14,15)(17,20,21,34,23,36,22,35)(18,25,26,42,27,43,37,47)(19,29,31,33,32,45,30,44)(24,38,28,41,40,46,39,48);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,36,35)(24,39,40,28)(25,42,43,47)(29,44,45,33)(38,48,46,41)Polycyclic presentation:
2^9|a^2=fi,b^2=h,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,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7088162);