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 | |
4224 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)(3,11)(4,8)(5,7)(6,10)(9,14)(12,16)(13,15)(17,18,23,28)(19,29,31,24)(20,22,27,25)(21,32,30,26)(33,34)(35,37)(36,45)(38,40)(39,42)(41,44)(43,47)(46,48);(1,3,6,5)(2,7,10,11)(4,12,14,13)(8,15,9,16)(17,19)(18,24)(20,21)(22,26)(23,31)(25,32)(27,30)(28,29)(33,35)(34,37)(36,46)(38,43)(39,44)(40,47)(41,42)(45,48);(1,4,6,14)(2,8,10,9)(3,12,5,13)(7,15,11,16)(17,20,23,27)(18,25,28,22)(19,21,31,30)(24,32,29,26)(33,36,39,38)(34,40,42,45)(35,43,44,46)(37,48,41,47);(1,4)(2,9)(3,12)(5,13)(6,14)(7,16)(8,10)(11,15)(17,20)(18,25)(19,30)(21,31)(22,28)(23,27)(24,26)(29,32)(33,37,39,41)(34,35,42,44)(36,47,38,48)(40,46,45,43);(1,5,6,3)(2,7,10,11)(4,13,14,12)(8,15,9,16)(17,21,23,30)(18,26,28,32)(19,20,31,27)(22,29,25,24)(33,35)(34,41)(36,43)(37,42)(38,46)(39,44)(40,47)(45,48);(1,2)(3,7)(4,8)(5,11)(6,10)(9,14)(12,15)(13,16)(17,22,23,25)(18,27,28,20)(19,26,31,32)(21,24,30,29)(33,38,39,36)(34,40,42,45)(35,43,44,46)(37,47,41,48)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627906);