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 | |
5925 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,14,13)(11,15,16,12)(17,18,21,23)(19,24,22,20);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,15,13,12)(17,19)(18,20)(21,22)(23,24);(2,4)(6,8)(10,13)(11,16)(17,20,21,24)(18,22,23,19);(3,7)(6,8)(9,10,14,13)(11,15,16,12)(17,20)(18,19)(21,24)(22,23);(1,3,5,7)(2,6,4,8)(9,12,14,15)(10,16,13,11)(17,19)(18,20)(21,22)(23,24);(1,4,5,2)(3,8,7,6)(9,13,14,10)(11,15,16,12)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7753833);