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 | |
4106 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(2,6)(7,9)(8,12)(15,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(2,6)(7,9)(8,12)(15,16)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31)(33,35,39,37)(34,41,36,40)(38,43,46,45)(42,48,44,47);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,14)(12,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,37,39,35)(34,40,36,41)(38,45,46,43)(42,47,44,48);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8648791);