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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,6,10)(4,5,9,15)(12,13,14,16)(17,18,22,26)(19,20,24,23)(21,29,31,25)(27,32,30,28)(33,34,39,44)(35,42,37,40)(36,46,47,41)(38,43,45,48);(1,3,7,6)(2,8,11,10)(4,12,9,14)(5,13,15,16)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,35,39,37)(34,40,44,42)(36,38,47,45)(41,43,46,48);(3,6)(4,9)(5,15)(8,10)(17,19,22,24)(18,23,26,20)(21,30,31,27)(25,28,29,32)(33,36,39,47)(34,41,44,46)(35,45,37,38)(40,48,42,43);(1,4)(2,5)(3,12)(6,14)(7,9)(8,13)(10,16)(11,15)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,37,39,35)(34,42,44,40)(36,38,47,45)(41,43,46,48);(1,5)(2,9)(3,13)(4,11)(6,16)(7,15)(8,14)(10,12)(17,20)(18,24)(19,26)(21,28)(22,23)(25,30)(27,29)(31,32)(33,38)(34,43)(35,36)(37,47)(39,45)(40,41)(42,46)(44,48);(1,6,7,3)(2,10,11,8)(4,14,9,12)(5,16,15,13)(17,21,22,31)(18,25,26,29)(19,27,24,30)(20,28,23,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=h,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8930786);