Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 | |
1709 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)(17,18,22,27,24,20,23,34)(19,28,31,29,32,41,39,25)(21,26,36,43,38,33,37,47)(30,42,35,44,45,46,48,40);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,25)(20,29)(21,35)(22,39)(23,31)(24,32)(26,42)(27,41)(28,34)(30,36)(33,46)(37,45)(38,48)(40,43)(44,47);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20,24,18)(19,29,32,25)(21,33,38,26)(22,34,23,27)(28,31,41,39)(30,44,45,40)(35,46,48,42)(36,47,37,43);(4,12)(7,16)(9,11)(13,15)(17,21,23,37,24,38,22,36)(18,26,34,47,20,33,27,43)(19,30,39,48,32,45,31,35)(25,40,41,46,29,44,28,42);(2,8)(3,10)(7,16)(9,11)Polycyclic presentation:
2^9|a^2=fh,c^2=i,d^2=fi,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=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6941823);