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:
(1,2,7,6)(3,9,4,8)(5,10,13,14)(11,16,12,15)(17,18,23,25)(19,24,28,22)(20,26,30,31)(21,27,29,32)(33,34,39,43)(35,44,42,38)(36,47,46,40)(37,41,45,48);(1,3,7,4)(2,8,6,9)(5,11,13,12)(10,15,14,16)(17,19,23,28)(18,24,25,22)(20,29,30,21)(26,32,31,27)(33,35,39,42)(34,38,43,44)(36,45,46,37)(40,48,47,41);(18,25)(19,28)(20,30)(27,32)(33,36,39,46)(34,40,43,47)(35,45,42,37)(38,48,44,41);(1,4,7,3)(2,9,6,8)(5,11,13,12)(10,15,14,16)(17,20)(18,26)(19,29)(21,28)(22,27)(23,30)(24,32)(25,31)(33,36)(34,40)(35,45)(37,42)(38,48)(39,46)(41,44)(43,47);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,21,23,29)(18,27,25,32)(19,30,28,20)(22,26,24,31)(33,37,39,45)(34,41,43,48)(35,46,42,36)(38,47,44,40);(1,6,7,2)(3,9,4,8)(5,10,13,14)(11,15,12,16)(17,22,23,24)(18,28,25,19)(20,27,30,32)(21,31,29,26)(33,38)(34,42)(35,43)(36,48)(37,40)(39,44)(41,46)(45,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=hi,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10050817);