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) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(4,12)(7,9)(10,16)(17,18,23,25)(19,24,22,32)(20,28,30,21)(26,31,29,27)(33,34,39,37)(35,42,38,40)(36,41,44,46)(43,48,45,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,23,22)(18,24,25,32)(20,29,30,26)(21,31,28,27)(34,37)(36,44)(40,42)(43,45);(1,2,5,6)(3,7,11,9)(4,8,12,13)(10,14,16,15)(17,20,23,30)(18,21,25,28)(19,26,22,29)(24,27,32,31)(33,35,39,38)(34,40,37,42)(36,45,44,43)(41,48,46,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20)(18,21)(19,26)(22,29)(23,30)(24,27)(25,28)(31,32)(33,36,39,44)(34,41,37,46)(35,43,38,45)(40,47,42,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,21)(18,20)(19,27)(22,31)(23,28)(24,26)(25,30)(29,32)(33,37,39,34)(35,40,38,42)(36,46,44,41)(43,47,45,48);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,22,23,19)(18,24,25,32)(20,29,30,26)(21,27,28,31)(33,38,39,35)(34,42,37,40)(36,45,44,43)(41,48,46,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9947904);