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 | |
4245 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18,22,23)(19,24,27,21)(20,25,30,31)(26,32,28,29)(33,34,38,36)(35,40,42,39)(37,44,45,41)(43,46,48,47);(2,5)(4,12)(7,9)(10,13)(18,23)(21,24)(25,31)(29,32)(35,42)(37,45)(39,40)(41,44);(1,3,6,11)(2,7,5,9)(4,13,12,10)(8,16,14,15)(17,19,22,27)(18,24,23,21)(20,28,30,26)(25,29,31,32)(33,35,38,42)(34,39,36,40)(37,43,45,48)(41,46,44,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,13)(14,16)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32)(33,36,38,34)(35,39,42,40)(37,44,45,41)(43,46,48,47);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,20)(18,25)(19,26)(21,29)(22,30)(23,31)(24,32)(27,28)(33,35)(34,40)(36,39)(37,43)(38,42)(41,47)(44,46)(45,48);(1,5,6,2)(3,9,11,7)(4,14,12,8)(10,16,13,15)(17,21)(18,19)(20,29)(22,24)(23,27)(25,26)(28,31)(30,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=gh,c^2=gi,d^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8725566);