Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6473167 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1634 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)(3,6)(4,11)(5,7)(8,12)(9,13)(10,16)(14,15)(17,18,23,27,24,28,22,26)(19,29,31,39,33,40,32,21)(20,34,37,25,38,41,36,42)(30,44,45,48,47,35,46,43);(1,3)(2,6)(4,10)(5,9)(7,13)(8,14)(11,16)(12,15)(17,19)(18,21)(20,30)(22,31)(23,32)(24,33)(25,35)(26,29)(27,40)(28,39)(34,43)(36,45)(37,46)(38,47)(41,48)(42,44);(2,7)(3,9)(8,11)(10,15)(17,20,23,37,24,38,22,36)(18,25,26,34,28,42,27,41)(19,30,32,46,33,47,31,45)(21,35,29,43,39,44,40,48);(3,9)(6,13)(10,15)(14,16)(17,21)(18,19)(20,35)(22,40)(23,29)(24,39)(25,30)(26,32)(27,31)(28,33)(34,46)(36,48)(37,43)(38,44)(41,45)(42,47);(1,4,5,12)(2,8,7,11)(3,10,9,15)(6,14,13,16)(17,22,24,23)(18,26,28,27)(19,31,33,32)(20,36,38,37)(21,29,39,40)(25,34,42,41)(30,45,47,46)(35,43,44,48)Polycyclic presentation:
2^9|a^2=f,c^2=gh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405882);