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 | |
7262 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)(17,18,22,23)(19,21,20,24);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,16)(14,15)(18,23)(19,20);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,20)(18,24,23,21);(2,6)(4,7)(9,11,12,16)(10,13,14,15)(17,20,22,19)(18,21,23,24);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,21)(18,19)(20,23)(22,24);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232278, #72232817, #72241230, … |
| maximal normal subgroup of | #72232278, #72232817, #72241230, … |
| maximal quotient of | #72232364, #72232660, #72232888, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064066);