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 | |
4156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18)(19,24)(20,22)(21,26)(23,25)(27,28)(29,32)(30,31)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,2,6,7)(3,9,12,8)(4,13,15,5)(10,16,14,11)(17,19,23,26)(18,24,25,21)(20,27,29,31)(22,28,32,30)(34,35)(39,41)(40,42)(46,47);(2,7)(5,13)(8,9)(11,16)(18,25)(19,26)(22,32)(27,31)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,3,6,12)(2,8,7,9)(4,14,15,10)(5,11,13,16)(17,20)(18,22)(19,27)(21,30)(23,29)(24,28)(25,32)(26,31)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4)(2,5)(3,10)(6,15)(7,13)(8,16)(9,11)(12,14)(17,21,23,24)(18,19,25,26)(20,28,29,30)(22,31,32,27)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,5)(2,4)(3,11)(6,13)(7,15)(8,14)(9,10)(12,16)(17,22)(18,20)(19,28)(21,31)(23,32)(24,27)(25,29)(26,30)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8300847);