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) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,8)(5,7)(6,13)(10,14)(11,15)(12,16)(17,18,23,26)(19,24,22,20)(21,29,31,25)(27,28,30,32)(33,34,39,41)(35,37,43,36)(38,40,47,46)(42,45,48,44);(2,5)(8,9)(10,13)(15,16)(17,19,23,22)(18,24,26,20)(21,30,31,27)(25,28,29,32)(33,35,39,43)(34,36,41,37)(38,42,47,48)(40,44,46,45);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,20)(18,22)(19,26)(21,28)(23,24)(25,30)(27,29)(31,32)(33,36,39,37)(34,35,41,43)(38,44,47,45)(40,42,46,48);(1,4,7,3)(2,9,5,8)(6,11,14,12)(10,15,13,16)(17,20)(18,19)(21,28)(22,26)(23,24)(25,27)(29,30)(31,32)(33,34)(35,36)(37,43)(38,46)(39,41)(40,47)(42,45)(44,48);(1,5,7,2)(3,9,4,8)(6,13,14,10)(11,16,12,15)(17,21,23,31)(18,25,26,29)(19,27,22,30)(20,28,24,32)(33,37,39,36)(34,35,41,43)(38,45,47,44)(40,42,46,48);(1,6,7,14)(2,10,5,13)(3,11,4,12)(8,15,9,16)(17,22,23,19)(18,24,26,20)(21,30,31,27)(25,32,29,28)(33,38)(34,40)(35,42)(36,44)(37,45)(39,47)(41,46)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=hi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10349728);