Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514148 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23906 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,17)(11,18,20,24)(12,15,21,16)(14,22,23,19);(2,5)(6,8)(11,19)(12,21)(14,18)(15,16)(20,22)(23,24);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,22)(13,20)(15,18)(16,24)(17,23)(19,21);(10,15)(12,21)(14,18)(16,17)(19,22)(23,24);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=i,c^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72213480, #72219568, #72219569 |
| maximal normal subgroup of | #72213480, #72219568, #72219569 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422073);