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 | |
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)(3,6,9,13)(4,11,12,7)(10,16,15,14)(18,24)(19,28)(21,22)(26,38)(27,35)(29,43)(30,31)(32,42)(33,36)(37,40)(44,47)(45,46);(3,9)(4,12)(6,13)(7,11)(17,18)(19,25)(20,32)(21,24)(22,35)(23,27)(26,33)(28,40)(29,44)(30,37)(31,41)(34,38)(36,42)(39,45)(43,48)(46,47);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19,23,31)(18,25,27,41)(20,29,34,46)(21,30,22,28)(24,37,35,40)(26,39,42,48)(32,44,38,47)(33,45,36,43);(3,9)(6,13)(10,15)(14,16)(17,20,23,34)(18,26,27,42)(19,29,31,46)(21,33,22,36)(24,38,35,32)(25,39,41,48)(28,43,30,45)(37,47,40,44);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,21,23,22)(18,24,27,35)(19,30,31,28)(20,33,34,36)(25,37,41,40)(26,38,42,32)(29,45,46,43)(39,47,48,44)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=i,e^2=hi,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,f^b=fi,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, 7073442);