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 | #2825 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3805 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18)(19,24)(20,28)(21,26)(22,30)(23,27)(25,31)(29,32)(33,34,39,36)(35,43,44,40)(37,41,47,45)(38,42,48,46);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(17,19,23,22)(18,24,27,30)(20,29,31,26)(21,28,32,25)(33,35)(34,40)(36,43)(37,38)(39,44)(41,46)(42,45)(47,48);(2,7)(4,10)(9,13)(12,14)(17,20,23,31)(18,25,27,28)(19,26,22,29)(21,24,32,30)(33,34,39,36)(35,40,44,43)(37,45,47,41)(38,42,48,46);(1,2)(3,10)(4,8)(5,9)(6,7)(11,14)(12,16)(13,15)(17,20)(18,25)(19,26)(21,30)(22,29)(23,31)(24,32)(27,28)(33,36,39,34)(35,40,44,43)(37,45,47,41)(38,46,48,42);(1,4,6,10)(2,8,7,3)(5,14,15,12)(9,11,13,16)(17,21)(18,26)(19,25)(20,30)(22,28)(23,32)(24,31)(27,29)(33,37,39,47)(34,41,36,45)(35,38,44,48)(40,46,43,42);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,22,23,19)(18,24,27,30)(20,29,31,26)(21,25,32,28)(33,38)(34,42)(35,37)(36,46)(39,48)(40,45)(41,43)(44,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9947931);