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 | |
4158 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)(3,4)(5,7)(6,8)(9,11)(10,12)(13,14)(15,16)(17,18,23,27)(19,24,28,22)(20,21,26,25)(29,31,32,30)(33,34)(35,40)(36,46)(37,41)(38,47)(39,42)(43,44)(45,48);(1,2,6,8)(3,4,9,11)(5,7,13,14)(10,12,15,16)(17,19,23,28)(18,24,27,22)(20,29,26,32)(21,31,25,30)(33,35,39,45)(34,40,42,48)(36,41,47,43)(37,38,44,46);(3,9)(4,11)(5,13)(7,14)(19,28)(20,26)(21,25)(22,24)(33,36,39,47)(34,38,42,46)(35,43,45,41)(37,48,44,40);(1,3,6,9)(2,4,8,11)(5,10,13,15)(7,12,14,16)(17,20)(18,25)(19,29)(21,27)(22,31)(23,26)(24,30)(28,32)(33,36)(34,38)(35,41)(37,48)(39,47)(40,44)(42,46)(43,45);(1,4)(2,3)(5,12)(6,11)(7,10)(8,9)(13,16)(14,15)(17,21,23,25)(18,26,27,20)(19,30,28,31)(22,32,24,29)(33,37,39,44)(34,41,42,43)(35,38,45,46)(36,40,47,48);(1,5,6,13)(2,7,8,14)(3,10,9,15)(4,12,11,16)(17,22)(18,19)(20,31)(21,32)(23,24)(25,29)(26,30)(27,28)(33,38)(34,36)(35,44)(37,45)(39,46)(40,41)(42,47)(43,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=h,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8961683);