Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501096 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #25773 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1088 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)(4,5)(6,8)(9,10)(11,13)(12,15)(14,18)(16,22)(17,20)(19,24)(21,23);(2,5)(6,7)(9,11,13,10)(12,19,20,23)(14,16,22,18)(15,21,17,24);(9,12)(10,15)(11,17)(13,20)(14,21,22,24)(16,23,18,19);(1,3,4,8)(2,6,5,7)(10,11)(15,17)(16,18)(19,23)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^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=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72316505, #72316519, #72316520, … |
| maximal normal subgroup of | #72316505, #72316519, #72316520, … |
| maximal quotient of | #72318273 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410021);