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 | |
5822 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)(9,10,13,11)(12,15,16,14)(18,22)(23,24);(1,3)(2,6)(4,7)(5,8)(17,18)(19,23)(20,24)(21,22);(1,2,5,4)(3,7,8,6)(10,11)(14,15)(17,19,21,20)(18,23,22,24);(1,3)(2,6)(4,7)(5,8)(9,11,13,10)(12,15,16,14)(17,20,21,19)(18,24,22,23);(1,4,5,2)(3,7,8,6)(9,12,13,16)(10,14,11,15);(1,3)(2,7)(4,6)(5,8)Polycyclic presentation:
2^9|a^2=hi,c^2=g,d^2=gh,e^2=h,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 | #72262917, #72262946, #72263022, … |
| maximal normal subgroup of | #72262917, #72262946, #72263022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7825409);