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 | |
6197 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)(11,12)(13,15)(14,16)(19,20)(22,24);(2,3)(6,7)(9,11,14,15)(10,13,16,12)(17,18)(19,24)(20,22)(21,23);(1,2,5,3)(4,6,8,7)(17,19,21,20)(18,22,23,24);(9,12,14,13)(10,15,16,11)(17,18)(19,24)(20,22)(21,23);(1,3,5,2)(4,6,8,7)(9,13,14,12)(10,15,16,11)(17,18)(19,22)(20,24)(21,23);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,15)(14,16)(17,20,21,19)(18,22,23,24)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72262860, #72262907, #72262919, … |
| maximal normal subgroup of | #72262860, #72262907, #72262919, … |
| maximal quotient of | #72300497, #72300522, #72300755, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9756989);