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 | |
4065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,6,9,15)(4,5,8,14)(11,12,13,16)(19,25)(22,28)(26,31)(29,32)(33,34,39,38)(35,43,45,40)(36,46,44,41)(37,47,48,42);(1,3)(2,6)(4,11)(5,12)(7,9)(8,13)(10,15)(14,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,35,39,45)(34,40,38,43)(36,37,44,48)(41,47,46,42);(3,9)(6,15)(11,13)(12,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,36,39,44)(34,41,38,46)(35,37,45,48)(40,47,43,42);(1,4)(2,5)(3,11)(6,12)(7,8)(9,13)(10,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35)(34,40)(36,37)(38,43)(39,45)(41,47)(42,46)(44,48);(1,5)(2,8)(3,12)(4,10)(6,13)(7,14)(9,16)(11,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,37)(34,42)(35,44)(36,45)(38,47)(39,48)(40,41)(43,46);(1,6,7,15)(2,9,10,3)(4,12,8,16)(5,13,14,11)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,38,39,34)(35,43,45,40)(36,41,44,46)(37,47,48,42)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8231412);