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 | |
6980 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)(11,15)(12,14)(13,16);(3,7)(6,8)(9,10,12,14)(11,13,16,15)(17,18,21,20)(19,23,24,22);(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,19,21,24)(18,22,20,23);(9,10,12,14)(11,13,16,15)(17,19,21,24)(18,23,20,22);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,22)(21,24);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,13,14,15)(17,20,21,18)(19,22,24,23)Polycyclic presentation:
2^9|b^2=gi,c^2=gi,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72249417, #72249472, #72260805, … |
| maximal normal subgroup of | #72249417, #72249472, #72260805, … |
| maximal quotient of | #72260839, #72263273, #72263282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8875332);