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 | |
4140 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)(3,9,10,8)(4,5,13,14)(11,15,16,12)(17,18)(19,24)(20,26)(21,27)(22,28)(23,25)(29,32)(30,31)(33,34,39,43)(35,42,46,40)(36,38,37,41)(44,45,47,48);(2,7)(3,10)(5,14)(11,16)(17,19)(18,24)(20,21)(22,25)(23,28)(26,27)(29,30)(31,32)(33,35)(34,40)(36,47)(37,44)(38,45)(39,46)(41,48)(42,43);(2,7)(4,13)(8,9)(11,16)(18,25)(19,28)(21,29)(26,31)(33,36,39,37)(34,41,43,38)(35,44,46,47)(40,45,42,48);(1,3,6,10)(2,8,7,9)(4,11,13,16)(5,12,14,15)(17,20)(18,26)(19,29)(21,28)(22,27)(23,30)(24,32)(25,31)(33,37,39,36)(34,38,43,41)(35,44,46,47)(40,45,42,48);(1,4)(2,5)(3,11)(6,13)(7,14)(8,12)(9,15)(10,16)(17,21,23,29)(18,27,25,32)(19,20,28,30)(22,31,24,26)(33,35)(34,42)(36,44)(37,47)(38,45)(39,46)(40,43)(41,48);(1,5)(2,4)(3,12)(6,14)(7,13)(8,11)(9,16)(10,15)(17,22)(18,19)(20,27)(21,31)(23,24)(25,28)(26,29)(30,32)(33,38)(34,36)(35,45)(37,43)(39,41)(40,47)(42,44)(46,48)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=g,e^2=gh,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, 10243991);