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 | |
4265 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)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(17,18)(19,20)(21,29)(22,23)(24,26)(25,30)(27,31)(28,32)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(2,7)(3,10)(8,13)(11,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,31)(30,32)(34,35)(39,41)(40,42)(46,47);(2,7)(4,9)(8,13)(12,15)(18,23)(19,26)(25,29)(27,32)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,15)(14,16)(17,20,22,24)(18,19,23,26)(21,28,30,31)(25,27,29,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,29)(30,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|c^2=ghi,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8025992);