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 | |
6164 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)(3,8)(4,6)(5,7)(9,10,13,12)(11,15,16,14)(17,18)(19,20)(21,23)(22,24);(1,3)(2,4)(5,6)(7,8)(9,10,13,12)(11,15,16,14)(17,19,22,21)(18,20,24,23);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,15)(13,16)(17,19,22,21)(18,23,24,20);(9,10,13,12)(11,15,16,14)(17,20,22,23)(18,19,24,21);(1,4,5,8)(2,6,7,3)(9,12,13,10)(11,15,16,14)(17,21,22,19)(18,23,24,20);(1,2)(3,4)(5,7)(6,8)(9,11)(10,15)(12,14)(13,16)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=ghi,d^2=h,e^2=gh,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
| maximal subgroup of | #72263262, #72304347, #72304427, … |
| maximal normal subgroup of | #72263262, #72304347, #72304427, … |
| maximal quotient of | #72263301 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7598692);