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 | |
4095 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,23,26)(19,27,29,22)(20,25,21,24)(28,31,30,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(2,7)(3,10)(9,12)(11,16)(17,19)(18,22)(20,28)(21,30)(23,29)(24,31)(25,32)(26,27)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20,23,21)(18,24,26,25)(19,28,29,30)(22,31,27,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,21,23,20)(18,25,26,24)(19,28,29,30)(22,31,27,32)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,47)(44,48);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,19)(18,22)(20,30)(21,28)(23,29)(24,32)(25,31)(26,27)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,22)(18,19)(20,31)(21,32)(23,27)(24,28)(25,30)(26,29)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gi,d^2=hi,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8488660);