Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470237 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(12,13)(15,18)(16,22)(19,20);(3,4)(6,7)(9,10)(11,18)(12,22)(13,16)(14,17)(15,21)(19,23)(20,24);(1,3,5,4)(2,6,8,7)(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24);(1,4,5,3)(2,7,8,6)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24);(2,8)(6,7)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)Polycyclic presentation:
2^9|c^2=h,d^2=h,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72252419, #72252429, #72252430, … |
| maximal normal subgroup of | #72252419, #72252429, #72252430, … |
| maximal quotient of | #72267155 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394849);