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 | |
6592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,6)(3,7,8,4)(9,10,13,14)(11,12,16,15)(18,21)(19,20);(1,3)(2,4)(5,8)(6,7)(10,14)(12,15)(17,18,22,21)(19,24,20,23);(9,11)(10,15)(12,14)(13,16)(17,19,22,20)(18,23,21,24);(2,6)(4,7)(9,12,13,15)(10,11,14,16)(17,20,22,19)(18,24,21,23);(1,3)(2,4)(5,8)(6,7)(17,21,22,18)(19,24,20,23);(1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72234954, #72262965, #72263019, … |
| maximal normal subgroup of | #72234954, #72262965, #72263019, … |
| maximal quotient of | #72258904, #72266262, #72266575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8363206);