Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499787 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #59089 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1331 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(11,17)(15,22)(18,23)(19,24);(2,5)(6,8)(9,10,14,12)(11,15,19,18)(13,20,21,16)(17,22,24,23);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24);(2,5)(6,8)(9,12,14,10)(11,18,19,15)(13,20,21,16)(17,23,24,22)Polycyclic presentation:
2^9|b^2=i,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fh,h^c=hi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72231754, #72251753, #72251939, … |
| maximal normal subgroup of | #72231754, #72251753, #72251939, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407417);