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 | |
3962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,7,11)(5,8,14,16)(10,12,13,15)(17,18)(19,20)(21,24)(22,30)(23,26)(25,32)(27,29)(28,31)(34,36)(35,42)(38,46)(45,47);(1,3)(2,4)(5,13)(6,7)(8,15)(9,11)(10,14)(12,16)(17,19,23,24)(18,20,26,21)(22,31,32,27)(25,29,30,28)(33,34,39,36)(35,37,42,40)(38,41,46,44)(43,45,48,47);(3,7)(4,11)(10,13)(12,15)(17,20,23,21)(18,24,26,19)(22,28,32,29)(25,31,30,27)(33,35,39,42)(34,40,36,37)(38,43,46,48)(41,47,44,45);(1,4,6,11)(2,7,9,3)(5,12,14,15)(8,13,16,10)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,36,39,34)(35,37,42,40)(38,44,46,41)(43,45,48,47);(1,2)(3,4)(5,8)(6,9)(7,11)(10,12)(13,15)(14,16)(17,21,23,20)(18,19,26,24)(22,29,32,28)(25,27,30,31)(33,37)(34,35)(36,42)(38,45)(39,40)(41,43)(44,48)(46,47);(1,5)(2,8)(3,10)(4,12)(6,14)(7,13)(9,16)(11,15)(17,22,23,32)(18,25,26,30)(19,27,24,31)(20,28,21,29)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8748617);