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 | |
4185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,5,4,11)(6,9,15,16)(10,13,12,14)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34)(35,43)(36,37)(38,40)(39,41)(42,47)(44,45)(46,48);(1,3)(2,5)(4,7)(6,10)(8,11)(9,13)(12,15)(14,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,32)(29,31)(33,35,39,38)(34,40,41,43)(36,46,42,44)(37,45,47,48);(2,8)(3,4)(9,16)(10,12)(20,28)(24,29)(26,32)(30,31)(34,41)(36,42)(40,43)(44,46);(1,4,7,3)(2,5,8,11)(6,10,15,12)(9,14,16,13)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,36,39,42)(34,37,41,47)(35,44,38,46)(40,48,43,45);(1,5)(2,3)(4,8)(6,14)(7,11)(9,12)(10,16)(13,15)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,32)(29,30)(33,37)(34,42)(35,45)(36,41)(38,48)(39,47)(40,44)(43,46);(1,6)(2,9)(3,10)(4,12)(5,13)(7,15)(8,16)(11,14)(17,21,22,18)(19,25,27,23)(20,24,28,29)(26,30,32,31)(33,38,39,35)(34,40,41,43)(36,44,42,46)(37,45,47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=h,f^2=g,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, 8641570);