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 | |
7828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(18,22)(19,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,24)(22,23);(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,16,15)(17,19)(18,20)(21,24)(22,23);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,23)(22,24);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)Polycyclic presentation:
2^9|d^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232069, #72232195, #72232817, … |
| maximal normal subgroup of | #72232069, #72232195, #72232817, … |
| maximal quotient of | #72232101, #72232134, #72232535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066505);