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 | |
1731 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)(17,18,21,27,23,28,22,26)(19,29,31,24,33,44,32,39)(20,25,35,42,37,43,36,41)(30,40,45,38,46,48,34,47);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19,23,33)(18,24,28,39)(20,34,37,45)(21,32,22,31)(25,40,43,48)(26,44,27,29)(30,35,46,36)(38,42,47,41);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,38)(26,29)(27,44)(28,39)(34,36)(35,45)(37,46)(40,41)(42,48)(43,47);(4,12)(7,16)(9,11)(13,15)(17,20,21,35,23,37,22,36)(18,25,27,42,28,43,26,41)(19,30,31,45,33,46,32,34)(24,38,44,48,39,47,29,40);(2,8)(3,10)(7,16)(9,11)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,39,44)(25,41,43,42)(30,45,46,34)(38,40,47,48)Polycyclic presentation:
2^9|a^2=fh,b^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, 6941836);