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 | |
5843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,16)(12,15)(13,14)(17,18,21,24)(19,22,23,20);(1,2,4,6)(3,5,8,7)(10,14)(15,16)(17,19,21,23)(18,22,24,20);(1,2)(3,5)(4,6)(7,8)(17,20)(18,23)(19,24)(21,22);(9,11,13,12)(10,15,14,16)(17,19,21,23)(18,22,24,20);(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,19)(21,22)(23,24);(1,3)(2,5)(4,8)(6,7)(9,12,13,11)(10,15,14,16)(17,18)(19,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=gh,f^2=g,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 | #72326660, #72326915, #72331617 |
| maximal normal subgroup of | #72326660, #72326915, #72331617 |
| maximal quotient of | #72326773, #72326927, #72326979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9694344);