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 | |
6021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,7)(3,8,4,5)(9,10)(11,13)(12,15)(14,16)(18,20)(22,24);(1,3,6,4)(2,5,7,8)(10,11)(14,15)(19,23)(22,24);(2,7)(5,8)(9,10,13,11)(12,14,16,15)(17,18,21,20)(19,22,23,24);(2,7)(3,4)(9,11,13,10)(12,15,16,14)(17,19,21,23)(18,22,20,24);(1,4,6,3)(2,8,7,5)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,24,23,22);(1,5,6,8)(2,3,7,4)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=ghi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72311673, #72342294, #72342317, … |
| maximal normal subgroup of | #72311673, #72342294, #72342317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8247060);