Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#447 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10)(11,18)(12,17)(13,15)(14,21)(16,19)(20,24)(22,23);(1,2,3,4)(5,6)(7,8)(10,15)(16,23)(17,21)(18,24);(6,8)(9,11)(10,16)(12,19)(13,20)(14,22)(15,23)(17,24)(18,21);(5,6)(7,8)(9,12,13,14)(10,17,15,21)(11,19,20,22)(16,24,23,18);(6,8)Polycyclic presentation:
2^9|b^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=eh95 conjugacy classes — view the full character table.
| maximal subgroup of | #72310624, #72310634, #72310647, … |
| maximal normal subgroup of | #72310624, #72310634, #72310647, … |
| maximal quotient of | #72311716, #72311966 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481130);