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 | |
4043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,13,15,8)(4,5,11,10)(6,14,16,12)(17,18,23,27)(19,24,28,30)(20,22,26,32)(21,25,31,29)(33,34,39,42)(35,40,45,37)(36,38,48,41)(43,47,46,44);(1,3,7,15)(2,8,9,13)(4,6,11,16)(5,12,10,14)(17,19)(18,24)(20,29)(21,22)(23,28)(25,26)(27,30)(31,32)(33,35,39,45)(34,40,42,37)(36,46,48,43)(38,44,41,47);(2,9)(5,10)(8,13)(12,14)(19,28)(21,31)(24,30)(25,29)(33,36,39,48)(34,41,42,38)(35,43,45,46)(37,47,40,44);(1,4)(2,10)(3,6)(5,9)(7,11)(8,14)(12,13)(15,16)(17,20)(18,22)(19,29)(21,24)(23,26)(25,28)(27,32)(30,31)(33,37)(34,35)(36,47)(38,46)(39,40)(41,43)(42,45)(44,48);(1,5,7,10)(2,11,9,4)(3,14,15,12)(6,13,16,8)(17,21)(18,25)(19,22)(20,30)(23,31)(24,26)(27,29)(28,32)(33,36)(34,38)(35,43)(37,44)(39,48)(40,47)(41,42)(45,46);(1,6)(2,12)(3,11)(4,15)(5,13)(7,16)(8,10)(9,14)(17,22)(18,26)(19,21)(20,27)(23,32)(24,25)(28,31)(29,30)(33,38)(34,36)(35,44)(37,43)(39,41)(40,46)(42,48)(45,47)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8973203);