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 | |
5916 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)(3,7)(4,5)(6,8)(9,10,13,15)(11,12,16,14)(18,21)(22,24);(1,3)(2,6)(4,7)(5,8)(17,18)(19,22)(20,21)(23,24);(1,2,5,4)(3,7,8,6)(9,11,13,16)(10,14,15,12);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15);(1,4,5,2)(3,7,8,6)(9,12)(10,11)(13,14)(15,16)(17,19,20,23)(18,22,21,24);(1,3)(2,7)(4,6)(5,8)Polycyclic presentation:
2^9|a^2=hi,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72232285, #72232446, #72232833, … |
| maximal normal subgroup of | #72232285, #72232446, #72232833, … |
| maximal quotient of | #72232376, #72232569, #72232932, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7825586);