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 | |
5727 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)(3,4)(5,7)(6,8)(9,10,13,16)(11,14,15,12)(17,18,21,19)(20,23,24,22);(1,3,6,7)(2,5,8,4)(9,11,13,15)(10,14,16,12)(18,19)(22,23);(9,12)(10,15)(11,16)(13,14)(17,19,21,18)(20,23,24,22);(1,4,6,5)(2,7,8,3)(9,11,13,15)(10,14,16,12)(17,20)(18,22)(19,23)(21,24);(1,5,6,4)(2,7,8,3)(9,12)(10,11)(13,14)(15,16);(1,2)(3,5)(4,7)(6,8)(9,10)(11,14)(12,15)(13,16)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=hi,d^2=gi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72256162, #72262890, #72262899, … |
| maximal normal subgroup of | #72256162, #72262890, #72262899, … |
| maximal quotient of | #72266257, #72266532, #72266559 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9782696);