Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10,13,15)(11,12,16,14)(17,18,23,24)(19,25,32,36)(20,26,34,30)(21,27,35,29)(22,28,39,40)(31,41,38,44)(33,42,47,46)(37,43,48,45);(1,3,4,7)(2,5,6,8)(9,11,13,16)(10,14,15,12)(18,24)(21,35)(25,36)(26,30)(28,40)(37,48)(41,44)(42,46);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,19)(18,25)(20,27)(21,30)(22,38)(23,32)(24,36)(26,35)(28,44)(29,34)(31,39)(33,45)(37,42)(40,41)(43,47)(46,48);(9,12)(10,11)(13,14)(15,16)(17,20,23,34)(18,26,24,30)(19,29,32,27)(21,36,35,25)(22,33,39,47)(28,42,40,46)(31,45,38,43)(37,44,48,41);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,21)(18,27)(19,30)(20,25)(22,37)(23,35)(24,29)(26,32)(28,43)(31,46)(33,41)(34,36)(38,42)(39,48)(40,45)(44,47);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,22)(18,28)(19,31)(20,33)(21,37)(23,39)(24,40)(25,41)(26,42)(27,43)(29,45)(30,46)(32,38)(34,47)(35,48)(36,44)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837692);