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 | |
4271 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,9,8)(5,13,14,10)(11,15,16,12)(17,18,22,24)(19,25,27,21)(20,23,28,31)(26,32,29,30)(33,34,39,42)(35,40,36,38)(37,41,46,48)(43,47,44,45);(2,7)(3,9)(5,14)(12,15)(17,19,22,27)(18,21,24,25)(20,26,28,29)(23,30,31,32)(35,36)(38,40)(43,44)(45,47);(2,7)(4,8)(10,13)(12,15)(20,28)(23,31)(26,29)(30,32)(33,35,39,36)(34,40,42,38)(37,44,46,43)(41,45,48,47);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,15,13,12)(17,19)(18,21)(20,29)(22,27)(23,32)(24,25)(26,28)(30,31)(33,36,39,35)(34,38,42,40)(37,43,46,44)(41,47,48,45);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,20,22,28)(18,23,24,31)(19,26,27,29)(21,30,25,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,21)(18,19)(20,30)(22,25)(23,26)(24,27)(28,32)(29,31)(33,38)(34,35)(36,42)(37,45)(39,40)(41,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=i,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10270866);