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 | |
6254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,13,16)(11,14,15,12)(17,18,20,23)(19,22,24,21);(1,3,5,4)(2,6,7,8)(9,11,13,15)(10,14,16,12)(17,18)(19,21)(20,23)(22,24);(9,11)(10,12)(13,15)(14,16)(17,19)(18,21)(20,24)(22,23);(2,7)(6,8)(9,11,13,15)(10,14,16,12)(17,19,20,24)(18,21,23,22);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23);(1,4,5,3)(2,6,7,8)(9,12,13,14)(10,15,16,11)(17,18)(19,21)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232575, #72232576, #72262845, … |
| maximal normal subgroup of | #72232575, #72232576, #72262845, … |
| maximal quotient of | #72232597, #72232604, #72326425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9996342);