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 | |
6670 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)(9,10,12,14)(11,15,16,13)(17,18,20,21)(19,23,24,22);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(18,21)(22,23);(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,21)(23,24);(2,6)(4,7)(10,14)(13,15)(17,18)(19,22)(20,21)(23,24);(1,4)(2,3)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,20,24)(18,22,21,23);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,16)(14,15)Polycyclic presentation:
2^9|a^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72262965, #72263006, #72263025, … |
| maximal normal subgroup of | #72262965, #72263006, #72263025, … |
| maximal quotient of | #72266532, #72326419, #72326713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8050354);