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, prime radical, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,7,8)(3,6,5,4)(9,10,13,12)(11,15,16,14)(18,19)(22,23);(1,3,7,5)(2,4,8,6)(17,18,21,19)(20,22,24,23);(1,4,7,6)(2,5,8,3)(9,11,13,16)(10,14,12,15)(17,19,21,18)(20,22,24,23);(1,3,7,5)(2,4,8,6)(9,12,13,10)(11,14,16,15)(17,20,21,24)(18,22,19,23);(1,5,7,3)(2,4,8,6);(1,6,7,4)(2,5,8,3)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
| derived subgroup of | #72268207, #72271400 |
| Fitting subgroup of | #72268207 |
| maximal subgroup of | #72268207, #72271388 |
| maximal normal subgroup of | #72268207, #72271388 |
| maximal quotient of | #72268208 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7603605);