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 | |
4150 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)(3,4)(5,9)(6,10)(7,8)(11,12)(13,15)(14,16)(17,18)(19,26)(20,23)(21,24)(22,28)(25,30)(27,31)(29,32)(33,34,38,41)(35,39,44,37)(36,40,46,48)(42,47,43,45);(1,2,6,10)(3,7,8,4)(5,9,14,16)(11,15,13,12)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,38,44)(34,39,41,37)(36,43,46,42)(40,45,48,47);(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,3)(2,7)(4,10)(5,13)(6,8)(9,12)(11,14)(15,16)(17,20,23,18)(19,26,21,24)(22,28,30,25)(27,32,29,31)(33,36)(34,40)(35,43)(37,47)(38,46)(39,45)(41,48)(42,44);(1,4,6,7)(2,8,10,3)(5,12,14,15)(9,13,16,11)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,35)(34,37)(36,42)(38,44)(39,41)(40,45)(43,46)(47,48);(1,5,6,14)(2,9,10,16)(3,11,8,13)(4,12,7,15)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,44)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=h,d^2=h,e^2=g,f^2=g,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, 10327383);