Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433077 |
| 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 | #3006 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1660 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)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(18,23)(19,27)(21,34)(25,36)(26,37)(28,42)(29,30)(31,46)(33,41)(35,39)(44,45)(47,48);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,18,21,23,22,26,34,37)(19,24,29,35,30,40,27,39)(20,31,33,46,32,25,41,36)(28,43,45,48,44,38,42,47);(3,9)(6,12)(10,15)(13,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,35)(25,38)(26,40)(27,34)(31,43)(32,44)(33,45)(36,47)(37,39)(41,42)(46,48);(2,7)(6,12)(8,14)(13,16)(17,20)(18,25)(19,28)(21,33)(22,32)(23,36)(24,38)(26,31)(27,42)(29,45)(30,44)(34,41)(35,47)(37,46)(39,48)(40,43);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,13,12,16)(20,32)(25,31)(28,44)(33,41)(36,46)(38,43)(42,45)(47,48)Polycyclic presentation:
2^9|b^2=f,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344030);