Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499831 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #32898 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,15)(11,14,18,23)(13,21,20,16)(17,22,24,19);(1,3,4,8)(2,5,6,7)(9,10)(11,17)(12,15)(13,21)(14,19)(16,20)(18,24)(22,23);(9,11)(10,14)(12,18)(13,22)(15,23)(16,24)(17,21)(19,20);(2,6)(5,7)(10,15)(14,23)(16,21)(17,24)Polycyclic presentation:
2^9|a^2=i,b^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72233180, #72233436, #72238456, … |
| maximal normal subgroup of | #72233180, #72233436, #72238456, … |
| maximal quotient of | #72233178, #72233430, #72233829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407235);