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 | |
4152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,13,7)(4,5,15,10)(11,16,14,12)(19,25)(20,28)(23,26)(24,29)(33,34,38,39)(35,40,43,36)(37,41,46,47)(42,48,45,44);(1,3,6,13)(2,7,8,9)(4,11,15,14)(5,12,10,16)(17,18)(19,26)(20,29)(21,22)(23,25)(24,28)(27,31)(30,32)(34,39)(36,40)(41,47)(44,48);(2,8)(5,10)(7,9)(12,16)(18,22)(23,26)(24,29)(31,32)(33,35,38,43)(34,40,39,36)(37,45,46,42)(41,44,47,48);(1,3)(2,9)(4,14)(5,12)(6,13)(7,8)(10,16)(11,15)(17,19)(18,23)(20,30)(21,25)(22,26)(24,32)(27,28)(29,31)(33,36)(34,35)(37,44)(38,40)(39,43)(41,42)(45,47)(46,48);(1,4)(2,10)(3,11)(5,8)(6,15)(7,16)(9,12)(13,14)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,30)(26,32)(33,34)(35,40)(36,43)(37,41)(38,39)(42,48)(44,45)(46,47);(1,5,6,10)(2,4,8,15)(3,12,13,16)(7,11,9,14)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,37,38,46)(34,41,39,47)(35,42,43,45)(36,44,40,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839947);