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 | |
4048 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)(3,6)(4,7)(5,10)(8,14)(9,11)(12,16)(13,15)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(1,2)(3,6)(4,9)(5,8)(7,11)(10,14)(12,16)(13,15)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,3)(4,11)(8,10)(12,13)(18,19)(23,25)(24,26)(30,31)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,3,6,2)(4,9,11,7)(5,10,14,8)(12,16,13,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4,6,11)(2,7,3,9)(5,13,14,12)(8,16,10,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,36,37,44)(34,39,40,43)(35,41,42,48)(38,45,46,47)Polycyclic presentation:
2^9|b^2=ghi,c^2=hi,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9993555);