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 | |
5657 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)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(18,22)(19,23);(1,3,5,7)(2,4,8,6)(9,11,14,13)(10,15,16,12)(17,18)(19,20)(21,22)(23,24);(1,3)(2,6)(4,8)(5,7)(17,18)(19,24)(20,23)(21,22);(1,3,5,7)(2,4,8,6)(9,12,14,15)(10,13,16,11)(17,19,21,23)(18,20,22,24);(1,3)(2,4)(5,7)(6,8)(9,13,14,11)(10,15,16,12)(17,20,21,24)(18,23,22,19);(1,4)(2,7)(3,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,20)(21,22)(23,24)Polycyclic presentation:
2^9|b^2=gh,d^2=gi,e^2=g,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 | #72256162, #72263011, #72263019, … |
| maximal normal subgroup of | #72256162, #72263011, #72263019, … |
| maximal quotient of | #72266260, #72266513, #72266576 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10024456);