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 | |
7078 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)(3,4,7,8)(17,18,21,22)(19,24,23,20);(9,10)(11,14)(12,13)(15,16)(18,22)(19,23);(2,6)(4,8)(10,13)(14,16)(17,19,21,23)(18,20,22,24);(2,6)(3,7)(11,15)(14,16)(17,20)(18,23)(19,22)(21,24);(1,3,5,7)(2,4,6,8)(9,11,12,15)(10,14,13,16);(1,4)(2,7)(3,6)(5,8)(9,10)(11,14)(12,13)(15,16)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72312724, #72342317, #72342320 |
| maximal normal subgroup of | #72312724, #72342317, #72342320 |
| maximal quotient of | #72335299, #72335370, #72336387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7557183);