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 | |
7440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,5)(6,8)(9,10)(11,12)(13,14)(15,16);(3,7)(6,8)(10,14)(11,16)(17,18)(19,23)(20,21)(22,24);(1,2)(3,6)(4,5)(7,8)(11,16)(12,15)(18,21)(19,24);(9,11,13,16)(10,15,14,12)(17,19,20,24)(18,22,21,23);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,24)(21,23);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|d^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72256218, #72256734, #72260805, … |
| maximal normal subgroup of | #72256218, #72256734, #72260805, … |
| maximal quotient of | #72256755, #72256785, #72260624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8969366);