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 | |
4078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,12)(6,7)(8,14)(9,10)(11,16)(13,15)(19,20)(23,24)(27,28)(30,31)(34,38)(35,41)(36,44)(46,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(17,18,22,26)(19,23,20,24)(21,25,29,32)(27,30,28,31)(33,34,37,38)(35,39,41,42)(36,40,44,45)(43,46,47,48);(2,7)(3,9)(8,12)(11,15)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,47)(45,48);(1,2)(3,10)(4,12)(5,9)(6,7)(8,14)(11,13)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(1,4,6,14)(2,8,7,12)(3,11,9,15)(5,13,10,16)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,47)(45,48);(1,5,6,10)(2,9,7,3)(4,13,14,16)(8,15,12,11)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|b^2=hi,c^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066439);