Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| 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 |
| prime radical | #3036 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18,22,28,24,29,23,42)(19,30,32,35,33,43,31,25)(20,34,36,44,38,48,37,26)(21,27,40,46,41,47,39,45);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19)(18,25)(20,27)(21,34)(22,31)(23,32)(24,33)(26,40)(28,43)(29,35)(30,42)(36,45)(37,46)(38,47)(39,44)(41,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,23,37,24,38,22,36)(18,26,28,34,29,44,42,48)(19,27,32,46,33,47,31,45)(21,35,39,30,41,25,40,43);(3,10)(6,9)(11,16)(14,15)(17,21,22,40,24,41,23,39)(18,27,28,46,29,47,42,45)(19,26,32,34,33,44,31,48)(20,35,36,43,38,25,37,30);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(17,22,24,23)(18,28,29,42)(19,31,33,32)(20,36,38,37)(21,39,41,40)(25,43,35,30)(26,34,44,48)(27,45,47,46)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=g,d^2=f,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=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093937);