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:
(1,2,5,9)(3,7,10,6)(4,12,13,8)(11,14,16,15)(17,18,22,26)(19,27,28,23)(20,29,30,24)(21,31,32,25)(34,40)(36,44)(38,41)(43,46);(3,10)(6,7)(11,16)(14,15)(17,19)(18,23)(20,25)(21,24)(22,28)(26,27)(29,32)(30,31)(33,34,39,40)(35,41,37,38)(36,42,44,47)(43,48,46,45);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,28)(29,30)(31,32)(33,35,39,37)(34,41,40,38)(36,43,44,46)(42,48,47,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,18)(19,27)(20,29)(21,25)(22,26)(23,28)(24,30)(31,32)(33,36)(34,42)(35,43)(37,46)(38,45)(39,44)(40,47)(41,48);(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,20,22,30)(18,24,26,29)(19,25,28,31)(21,27,32,23)(33,37,39,35)(34,38,40,41)(36,43,44,46)(42,48,47,45);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,21,22,32)(18,25,26,31)(19,24,28,29)(20,27,30,23)(33,38)(34,35)(36,45)(37,40)(39,41)(42,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8046663);