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 | |
6710 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)(3,7)(4,6)(5,8)(10,14)(11,16)(18,21)(19,23);(9,10,13,14)(11,15,16,12)(18,21)(22,24);(1,2,4,6)(3,5,8,7)(9,11)(10,15)(12,14)(13,16)(17,18,20,21)(19,22,23,24);(17,19)(18,22)(20,23)(21,24);(1,2)(3,7)(4,6)(5,8)(9,12)(10,11)(13,15)(14,16);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,14)(13,16)Polycyclic presentation:
2^9|b^2=g,c^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72233523, #72233537, #72233561, … |
| maximal normal subgroup of | #72233523, #72233537, #72233561, … |
| maximal quotient of | #72261974, #72263208 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7571802);