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 | |
4116 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,18,23,27)(19,24,28,32)(20,22,26,31)(21,30,29,25)(34,40)(36,44)(37,45)(41,48);(3,4)(5,13)(7,10)(8,15)(19,28)(21,29)(24,32)(25,30)(33,34)(35,41)(36,42)(37,46)(38,44)(39,40)(43,45)(47,48);(1,3,6,4)(2,7,9,10)(5,11,13,12)(8,14,15,16)(17,19,23,28)(18,24,27,32)(20,29,26,21)(22,25,31,30)(33,35,39,47)(34,41,40,48)(36,45,44,37)(38,46,42,43);(1,4,6,3)(2,7,9,10)(5,12,13,11)(8,14,15,16)(17,20)(18,22)(19,29)(21,28)(23,26)(24,25)(27,31)(30,32)(33,36)(34,42)(35,45)(37,47)(38,40)(39,44)(41,43)(46,48);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,21,23,29)(18,25,27,30)(19,20,28,26)(22,24,31,32)(33,37,39,45)(34,43,40,46)(35,36,47,44)(38,48,42,41);(1,2)(3,7)(4,10)(5,8)(6,9)(11,14)(12,16)(13,15)(17,22)(18,26)(19,25)(20,27)(21,24)(23,31)(28,30)(29,32)(33,38,39,42)(34,44,40,36)(35,46,47,43)(37,48,45,41)Polycyclic presentation:
2^9|a^2=hi,c^2=gi,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8400418);