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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,8,4,7)(5,9,13,16)(11,15,12,14)(17,18)(19,24)(20,29)(21,25)(22,28)(23,26)(27,31)(30,32)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,48,44,41);(1,3,6,4)(2,7,10,8)(5,11,13,12)(9,14,16,15)(17,19)(18,24)(20,30)(21,27)(22,26)(23,28)(25,31)(29,32)(33,34)(35,39)(36,40)(37,48)(38,42)(41,44)(43,46)(45,47);(1,3,6,4)(2,8,10,7)(5,11,13,12)(9,15,16,14)(17,20,23,21)(18,25,26,29)(19,27,28,30)(22,31,24,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,41)(42,43)(47,48);(1,4,6,3)(2,8,10,7)(5,11,13,12)(9,14,16,15)(17,21,23,20)(18,25,26,29)(19,27,28,30)(22,32,24,31)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,41,43,48);(1,2)(3,8)(4,7)(5,9)(6,10)(11,15)(12,14)(13,16)(17,22)(18,19)(20,31)(21,32)(23,24)(25,27)(26,28)(29,30)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,40,43,47);(1,5,6,13)(2,9,10,16)(3,11,4,12)(7,14,8,15)(17,19)(18,22)(20,27)(21,30)(23,28)(24,26)(25,31)(29,32)(33,34)(35,39)(36,47)(37,41)(38,42)(40,45)(43,46)(44,48)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gh,d^2=gi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10358842);