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 | |
4173 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18,21,23)(19,22,25,30)(20,24,27,31)(26,29,28,32)(33,34,38,41)(35,37,43,39)(36,40,44,47)(42,45,48,46);(2,7)(4,15)(8,11)(10,13)(17,19,21,25)(18,22,23,30)(20,26,27,28)(24,29,31,32)(36,44)(40,47)(42,48)(45,46);(2,7)(3,12)(4,15)(5,16)(18,23)(19,25)(20,27)(29,32)(33,35,38,43)(34,39,41,37)(36,42,44,48)(40,46,47,45);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20)(18,24)(19,26)(21,27)(22,29)(23,31)(25,28)(30,32)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,19)(18,22)(20,28)(21,25)(23,30)(24,32)(26,27)(29,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,18)(19,22)(20,24)(21,23)(25,30)(26,29)(27,31)(28,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=ghi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9401939);