Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6473174 |
| 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 |
| prime radical | #60919 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1618 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)(17,18,24,29,23,30,22,28)(19,31,34,25,35,48,33,43)(20,36,41,44,37,45,38,26)(21,27,42,32,39,46,40,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,14)(11,16)(12,15)(17,19)(18,25)(20,32)(21,26)(22,34)(23,35)(24,33)(27,41)(28,48)(29,31)(30,43)(36,40)(37,47)(38,46)(39,44)(42,45);(2,7)(3,9)(8,11)(10,15)(17,20,22,38,23,37,24,41)(18,26,29,36,30,44,28,45)(19,32,34,46,35,47,33,27)(21,31,40,43,39,48,42,25);(3,9)(6,13)(10,15)(14,16)(17,21)(18,27)(19,26)(20,31)(22,40)(23,39)(24,42)(25,41)(28,47)(29,32)(30,46)(33,45)(34,36)(35,44)(37,48)(38,43);(1,4,5,12)(2,8,7,11)(3,10,9,15)(6,14,13,16)(20,37)(21,39)(26,44)(27,46)(32,47)(36,45)(38,41)(40,42)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,e^2=h,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^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405885);