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 | |
5707 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)(3,8)(4,6)(5,7)(10,13)(11,15)(17,18)(19,20)(21,24)(22,23);(1,3,5,4)(2,6,7,8)(9,10,14,13)(11,12,15,16)(17,18)(19,22)(20,23)(21,24);(9,11)(10,12)(13,16)(14,15)(17,18,21,24)(19,20,23,22);(9,12)(10,15)(11,13)(14,16)(17,19)(18,22)(20,24)(21,23);(1,2)(3,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24);(1,4,5,3)(2,6,7,8)(9,13,14,10)(11,16,15,12)(17,20)(18,23)(19,24)(21,22)Polycyclic presentation:
2^9|b^2=g,c^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72297123 |
| maximal normal subgroup of | #72297123 |
| maximal quotient of | #72297157, #72297554, #72297751 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9626501);