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 | |
4045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18)(19,26)(20,24)(21,29)(22,23)(25,31)(27,28)(30,32)(33,34)(35,37)(36,40)(38,42)(39,46)(41,47)(43,48)(44,45);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(18,23)(20,26)(25,29)(28,30)(33,35,38,41)(34,37,42,47)(36,44,46,43)(39,48,40,45);(2,6)(7,9)(8,12)(14,15)(17,19,22,24)(18,20,23,26)(21,27,31,32)(25,28,29,30)(33,36,38,46)(34,39,42,40)(35,43,41,44)(37,45,47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,30)(31,32)(33,36)(34,40)(35,44)(37,45)(38,46)(39,42)(41,43)(47,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20)(18,24)(19,23)(21,30)(22,26)(25,27)(28,31)(29,32)(33,35)(34,37)(36,43)(38,41)(39,45)(40,48)(42,47)(44,46);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,21,22,31)(18,25,23,29)(19,27,24,32)(20,28,26,30)(33,37)(34,41)(35,42)(36,45)(38,47)(39,44)(40,43)(46,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gi,e^2=ghi,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^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10066407);