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 | |
4016 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)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,34,39,45)(35,46,47,40)(36,42,48,41)(37,43,44,38);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(19,20)(23,26)(27,28)(31,32)(33,35)(34,40)(36,38)(37,41)(39,47)(42,44)(43,48)(45,46);(2,8)(4,9)(7,14)(10,15)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,36,39,48)(34,41,45,42)(35,38,47,43)(37,46,44,40);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,36)(34,42)(35,43)(37,40)(38,47)(39,48)(41,45)(44,46);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,37)(34,43)(35,42)(36,40)(38,45)(39,44)(41,47)(46,48);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,43)(34,44,45,37)(35,36,47,48)(40,42,46,41)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gi,d^2=ghi,f^2=g,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, 9952917);