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 | |
4072 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)(3,4)(5,7)(6,13)(8,10)(9,14)(11,16)(12,15)(17,18,23,21)(19,27,30,24)(20,31,29,25)(22,32,28,26)(33,34,38,41)(35,42,39,36)(37,45,46,40)(43,44,47,48);(1,3,7,8)(2,4,5,10)(6,11,14,15)(9,12,13,16)(17,19)(18,24)(20,28)(21,27)(22,29)(23,30)(25,26)(31,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,4,7,10)(2,8,5,3)(6,12,14,16)(9,15,13,11)(17,18,23,21)(19,24,30,27)(20,31,29,25)(22,26,28,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,4)(2,3)(5,8)(6,12)(7,10)(9,11)(13,15)(14,16)(17,20,23,29)(18,25,21,31)(19,28,30,22)(24,26,27,32)(33,36,38,42)(34,39,41,35)(37,44,46,48)(40,47,45,43);(1,5,7,2)(3,10,8,4)(6,13,14,9)(11,16,15,12)(17,21,23,18)(19,27,30,24)(20,25,29,31)(22,32,28,26)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,6,7,14)(2,9,5,13)(3,11,8,15)(4,12,10,16)(17,22,23,28)(18,26,21,32)(19,29,30,20)(24,25,27,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,d^2=g,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, 8066238);