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 | |
4122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(11,16)(14,15)(17,18,23,21)(19,26,24,20)(22,25,31,29)(27,30,32,28)(33,34,39,43)(35,44,38,40)(36,41,42,37)(45,48,47,46);(1,2)(3,7)(4,13)(5,9)(6,10)(8,12)(11,14)(15,16)(18,21)(20,26)(25,29)(28,30)(33,35,39,38)(34,40,43,44)(36,45,42,47)(37,48,41,46);(1,3)(2,5)(4,11)(6,9)(7,10)(8,14)(12,16)(13,15)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,30)(31,32)(33,36,39,42)(34,41,43,37)(35,45,38,47)(40,46,44,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,16)(13,14)(17,20,23,26)(18,24,21,19)(22,30,31,28)(25,27,29,32)(33,34)(35,40)(36,41)(37,42)(38,44)(39,43)(45,46)(47,48);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,21,23,18)(19,26,24,20)(22,29,31,25)(27,30,32,28)(33,37,39,41)(34,42,43,36)(35,46,38,48)(40,45,44,47);(1,5,6,7)(2,9,10,3)(4,14,12,15)(8,16,13,11)(17,22,23,31)(18,25,21,29)(19,27,24,32)(20,28,26,30)(33,38,39,35)(34,40,43,44)(36,47,42,45)(37,48,41,46)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=gi,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, 8404008);