Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470019 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4222 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)(10,15)(12,13)(16,24)(17,21)(18,23)(19,22);(3,4)(6,7)(9,10)(11,18)(12,15)(13,21)(14,17)(16,20)(19,24)(22,23);(1,3,5,4)(2,6,8,7)(9,11,14,20)(10,16,17,18)(12,19,13,22)(15,23,21,24);(1,4,5,3)(2,7,8,6)(10,17)(15,21)(16,18)(23,24);(2,8)(6,7)(9,12,14,13)(10,15,17,21)(11,19,20,22)(16,23,18,24)Polycyclic presentation:
2^9|c^2=h,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257908, #72308524, #72308533, … |
| maximal normal subgroup of | #72257908, #72308524, #72308533, … |
| maximal quotient of | #72264501, #72302866 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394688);