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 | |
7216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,7)(3,4,8,6)(9,10,13,15)(11,14,16,12)(18,21)(19,23);(1,3)(2,6)(4,7)(5,8)(9,11,13,16)(10,14,15,12)(17,18,20,21)(19,22,23,24);(2,7)(4,6)(10,15)(11,16)(17,18)(19,22)(20,21)(23,24);(2,7)(3,8)(9,12,13,14)(10,11,15,16)(17,18)(19,22)(20,21)(23,24);(1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,19)(18,22)(20,23)(21,24);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72232575, #72241094, #72241101, … |
| maximal normal subgroup of | #72232575, #72241094, #72241101, … |
| maximal quotient of | #72232624, #72293916, #72293917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8502545);