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 | |
4057 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)(3,8)(4,6)(5,9)(7,11)(10,15)(12,16)(13,14)(33,34,39,43)(35,44,38,40)(36,42,37,41)(45,47,46,48);(1,3,7,15)(2,8,11,10)(4,13,5,12)(6,14,9,16)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32)(33,35,39,38)(34,40,43,44)(36,46,37,45)(41,48,42,47);(1,4,7,5)(2,9,11,6)(3,12,15,13)(8,14,10,16)(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32)(33,34)(35,40)(36,41)(37,42)(38,44)(39,43)(45,47)(46,48);(1,5,7,4)(2,9,11,6)(3,13,15,12)(8,14,10,16)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32)(33,36,39,37)(34,41,43,42)(35,45,38,46)(40,47,44,48);(1,3)(2,10)(4,12)(5,13)(6,14)(7,15)(8,11)(9,16)(17,21,23,30)(18,25,22,29)(19,27,20,28)(24,31,26,32)(33,37,39,36)(34,42,43,41)(35,46,38,45)(40,48,44,47);(1,6,7,9)(2,4,11,5)(3,14,15,16)(8,13,10,12)(17,22,23,18)(19,24,20,26)(21,29,30,25)(27,31,28,32)(33,38,39,35)(34,40,43,44)(36,46,37,45)(41,47,42,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=i,d^2=gi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8177740);