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 | |
4364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,7)(4,9)(5,10)(6,11)(8,13)(12,15)(14,16)(17,18)(19,22)(20,27)(21,24)(23,28)(25,31)(26,30)(29,32)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,3)(2,7)(4,10)(5,9)(6,13)(8,11)(12,14)(15,16)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(34,40)(35,37)(42,46)(43,45);(1,3,6,13)(2,7,11,8)(4,12,14,10)(5,9,15,16)(17,18,21,24)(19,22,26,30)(20,23,28,27)(25,29,32,31)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,3)(2,8)(4,12)(5,9)(6,13)(7,11)(10,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,4)(2,9)(3,12)(5,8)(6,14)(7,15)(10,13)(11,16)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,5)(2,10)(3,9)(4,7)(6,15)(8,14)(11,12)(13,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)(33,38,39,47)(34,42,40,46)(35,43,37,45)(36,44,41,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=ghi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9877276);