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 | |
4240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,7)(3,9,11,6)(4,8,13,15)(10,16,12,14)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,3,5,11)(2,6,7,9)(4,10,13,12)(8,14,15,16)(17,18,22,23)(19,24,29,32)(20,25,30,28)(21,26,31,27)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(2,7)(6,9)(8,15)(14,16)(18,23)(20,30)(24,32)(26,27)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,3)(2,6)(4,12)(5,11)(7,9)(8,16)(10,13)(14,15)(17,19)(18,24)(20,27)(21,25)(22,29)(23,32)(26,30)(28,31)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,47)(44,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,20)(18,25)(19,27)(21,24)(22,30)(23,28)(26,29)(31,32)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21)(18,26)(19,28)(20,24)(22,31)(23,27)(25,29)(30,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8424700);