Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021351 |
| 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 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
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)(17,18,23,28,24,29,22,27)(19,30,33,42,35,45,34,25)(20,36,21,26,38,43,37,44)(31,47,48,41,32,40,39,46);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,25)(20,31)(21,39)(22,33)(23,34)(24,35)(26,40)(27,30)(28,45)(29,42)(32,38)(36,46)(37,48)(41,43)(44,47);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20,22,37,24,38,23,21)(18,26,28,43,29,44,27,36)(19,31,33,48,35,32,34,39)(25,40,45,41,42,47,30,46);(4,12)(7,16)(9,11)(13,15)(17,21,23,38,24,37,22,20)(18,26,28,43,29,44,27,36)(19,32,33,39,35,31,34,48)(25,41,30,40,42,46,45,47);(2,8)(3,10)(7,16)(9,11)(17,22,24,23)(18,27,29,28)(19,33,35,34)(20,37,38,21)(25,30,42,45)(26,36,44,43)(31,48,32,39)(40,46,47,41)Polycyclic presentation:
2^9|a^2=fh,c^2=gi,d^2=f,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=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949271);