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 | |
6342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,4)(3,7,8,6)(10,13)(14,15)(17,18,21,24)(19,22,23,20);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23);(9,10)(11,15)(12,13)(14,16)(17,18)(19,20)(21,24)(22,23);(9,11)(10,14)(12,16)(13,15)(17,20)(18,23)(19,24)(21,22);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,16)(13,14)(17,20)(18,19)(21,22)(23,24);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=gh,e^2=ghi,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 | #72232288, #72258018, #72285851, … |
| maximal normal subgroup of | #72232288, #72258018, #72285851, … |
| maximal quotient of | #72232325, #72297245, #72297249, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9622674);