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 | |
5117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(6,8)(9,10,15,13)(11,12,16,14);(3,7)(6,8)(10,13)(11,16)(17,18,22,23)(19,27,28,24)(20,25,29,31)(21,26,32,30);(1,2,5,4)(3,6,7,8)(9,11)(10,14)(12,13)(15,16)(18,23)(20,29)(24,27)(26,30);(9,12,15,14)(10,11,13,16)(17,19)(18,24)(20,30)(21,31)(22,28)(23,27)(25,32)(26,29);(1,3,5,7)(2,6,4,8)(9,13,15,10)(11,12,16,14)(17,20)(18,25)(19,26)(21,24)(22,29)(23,31)(27,32)(28,30);(1,4,5,2)(3,8,7,6)(9,14,15,12)(10,16,13,11)(17,21)(18,26)(19,25)(20,24)(22,32)(23,30)(27,29)(28,31)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72280763 |
| maximal normal subgroup of | #72280763 |
| maximal quotient of | #72280903 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8997249);