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 | |
5907 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,12,14)(11,13,15,16)(17,18,24,27)(19,28,30,25)(20,23,21,31)(22,26,32,29);(1,3,4,8)(2,5,6,7)(11,15)(13,16)(17,19)(18,25)(20,26)(21,29)(22,23)(24,30)(27,28)(31,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,20,24,21)(18,23,27,31)(19,26,30,29)(22,28,32,25);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,21,24,20)(18,23,27,31)(19,29,30,26)(22,28,32,25);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,22)(18,26)(19,23)(20,25)(21,28)(24,32)(27,29)(30,31);(17,23)(18,21)(19,22)(20,27)(24,31)(25,29)(26,28)(30,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=gi,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72330757, #72342303, #72342323, … |
| maximal normal subgroup of | #72330757, #72342303, #72342323, … |
| maximal quotient of | #72330931, #72331913, #72332057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8439322);