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)(17,18,21,25,22,27,35,26)(19,28,31,40,32,39,30,23)(20,24,36,42,37,34,48,43)(29,41,33,44,46,47,45,38);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,23)(20,33)(21,30)(22,32)(24,41)(25,39)(26,28)(27,40)(29,36)(31,35)(34,47)(37,45)(38,42)(43,44)(46,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,18)(19,23)(20,34)(21,25)(22,27)(24,37)(26,35)(28,31)(29,44)(30,39)(32,40)(33,47)(36,43)(38,46)(41,45)(42,48);(4,12)(7,16)(9,11)(13,15)(17,20,35,48,22,37,21,36)(18,24,26,43,27,34,25,42)(19,29,30,45,32,46,31,33)(23,38,39,47,40,44,28,41);(2,8)(3,10)(7,16)(9,11)(17,21,22,35)(18,25,27,26)(19,30,32,31)(20,36,37,48)(23,39,40,28)(24,42,34,43)(29,45,46,33)(38,47,44,41)Polycyclic presentation:
2^9|a^2=fh,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, 6941835);