Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 | |
1696 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)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18,23,27)(19,29,33,48)(20,34,37,45)(21,26,31,42)(22,40,30,28)(24,41,43,32)(25,35,44,36)(38,47,39,46);(3,9)(4,12)(6,13)(7,11)(17,19,30,41,23,33,22,32)(18,24,40,48,27,43,28,29)(20,31,36,38,37,21,35,39)(25,42,34,46,44,26,45,47);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20)(18,25)(19,31)(21,33)(22,35)(23,37)(24,42)(26,43)(27,44)(28,45)(29,47)(30,36)(32,39)(34,40)(38,41)(46,48);(3,9)(6,13)(10,15)(14,16)(17,21,23,31)(18,26,27,42)(19,20,33,37)(22,39,30,38)(24,25,43,44)(28,47,40,46)(29,34,48,45)(32,36,41,35);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,27)(24,43)(25,44)(26,42)(28,40)(29,48)(34,45)(46,47)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7074353);