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 | |
4157 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,9,4)(5,13,14,10)(11,12,16,15)(20,28)(23,30)(26,32)(29,31)(33,34)(35,41)(36,39)(37,40)(38,44)(42,46)(43,47)(45,48);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(34,36)(40,41)(42,44)(47,48);(2,8)(3,9)(10,13)(11,16)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,3)(2,7)(4,8)(5,11)(6,9)(10,15)(12,13)(14,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=hi,d^2=g,e^2=h,f^2=h,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, 8065226);