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 | |
4047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(7,14)(10,16)(13,15)(17,18)(19,28)(20,24)(21,29)(22,27)(23,30)(25,31)(26,32)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,19)(18,23)(20,29)(21,25)(22,30)(24,32)(26,31)(27,28)(34,40)(36,41)(42,46)(44,48);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,20,22,31)(18,24,27,25)(19,29,30,26)(21,23,32,28)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20)(18,25)(19,29)(21,23)(22,31)(24,27)(26,30)(28,32)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18)(19,23)(20,24)(21,26)(22,27)(25,31)(28,30)(29,32)(33,37,39,35)(34,36,40,41)(38,45,47,43)(42,44,46,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,21,22,32)(18,26,27,29)(19,25,30,24)(20,23,31,28)(33,38,39,47)(34,42,40,46)(35,43,37,45)(36,44,41,48)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064493);