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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1689 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)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18,22,27,23,28,21,26)(19,29,31,41,32,40,38,24)(20,25,36,44,37,34,35,43)(30,42,33,45,46,47,48,39);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19,23,32)(18,24,28,41)(20,33,37,48)(21,31,22,38)(25,42,34,47)(26,29,27,40)(30,35,46,36)(39,43,45,44);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,18,23,28)(19,24,32,41)(20,34,37,25)(21,26,22,27)(29,38,40,31)(30,45,46,39)(33,47,48,42)(35,44,36,43);(4,12)(7,16)(9,11)(13,15)(17,20,21,35,23,37,22,36)(18,25,26,43,28,34,27,44)(19,30,38,48,32,46,31,33)(24,39,40,47,41,45,29,42);(2,8)(3,10)(7,16)(9,11)(17,21,23,22)(18,26,28,27)(19,31,32,38)(20,35,37,36)(24,29,41,40)(25,43,34,44)(30,33,46,48)(39,42,45,47)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=i,d^2=fi,e^2=i,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=dhi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6941838);