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 | |
4136 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(17,18,23,21)(19,20,27,24)(22,29,30,25)(26,31,32,28)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,19,23,27)(18,24,21,20)(22,26,30,32)(25,31,29,28)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(2,6)(3,5)(4,12)(15,16)(18,21)(20,24)(22,30)(26,32)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,20)(18,19)(21,27)(22,28)(23,24)(25,26)(29,32)(30,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,5,7,3)(2,10,6,8)(4,11,12,14)(9,15,13,16)(17,21,23,18)(19,24,27,20)(22,29,30,25)(26,31,32,28)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,22)(18,25)(19,26)(20,28)(21,29)(23,30)(24,31)(27,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=ghi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,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, 9405571);