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 | |
4100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,18,23,20)(19,24,26,22)(21,25,31,28)(27,32,29,30)(33,34,39,42)(35,40,44,48)(36,45,43,41)(37,47,46,38);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,30,28,32)(33,35,39,44)(34,40,42,48)(36,46,43,37)(38,41,47,45);(4,12)(7,14)(10,16)(13,15)(19,26)(22,24)(27,29)(30,32)(33,36,39,43)(34,41,42,45)(35,37,44,46)(38,48,47,40);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20,23,18)(19,22,26,24)(21,25,31,28)(27,32,29,30)(33,35)(34,40)(36,37)(38,41)(39,44)(42,48)(43,46)(45,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37)(34,38)(35,43)(36,44)(39,46)(40,45)(41,48)(42,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,22)(18,19)(20,26)(21,30)(23,24)(25,27)(28,29)(31,32)(33,38)(34,37)(35,41)(36,40)(39,47)(42,46)(43,48)(44,45)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8267149);