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 | |
6273 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,8)(3,7,4,5)(9,10,13,12)(11,15,16,14)(17,18,21,20)(19,22,24,23);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,21)(22,24);(2,8)(3,4)(9,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,21)(22,24);(1,3,6,4)(2,5,8,7)(9,10,13,12)(11,15,16,14)(17,19)(18,22)(20,23)(21,24);(1,4,6,3)(2,5,8,7)(9,12,13,10)(11,15,16,14)(17,20,21,18)(19,23,24,22);(1,5)(2,4)(3,8)(6,7)(9,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,22)(21,24)Polycyclic presentation:
2^9|a^2=gi,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72262970, #72263013, #72263035, … |
| maximal normal subgroup of | #72262970, #72263013, #72263035, … |
| maximal quotient of | #72263512, #72278602, #72278611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9963007);