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 | |
4142 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)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,34)(35,40)(36,46)(37,41)(38,47)(39,42)(43,44)(45,48);(1,2,6,10)(3,7,8,4)(5,9,13,16)(11,14,15,12)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,35,39,45)(34,40,42,48)(36,41,47,43)(37,38,44,46);(3,8)(4,7)(5,13)(9,16)(17,18,22,25)(19,23,20,26)(21,29,30,24)(27,32,28,31)(33,36,39,47)(34,38,42,46)(35,43,45,41)(37,48,44,40);(1,3)(2,7)(4,10)(5,11)(6,8)(9,14)(12,16)(13,15)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,36)(34,38)(35,41)(37,48)(39,47)(40,44)(42,46)(43,45);(1,4,6,7)(2,8,10,3)(5,12,13,14)(9,15,16,11)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,37,39,44)(34,41,42,43)(35,38,45,46)(36,40,47,48);(1,5,6,13)(2,9,10,16)(3,11,8,15)(4,12,7,14)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)(33,38)(34,36)(35,44)(37,45)(39,46)(40,41)(42,47)(43,48)Polycyclic presentation:
2^9|b^2=gi,c^2=h,e^2=gi,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, 8961143);