Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511779 |
| 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 | C2×#560 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1037 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,13,17)(11,18,12,15)(14,16,21,24)(19,22,20,23);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,15,17,18)(14,19,21,20)(16,23,24,22);(10,16)(11,12)(15,22)(17,24)(18,23)(19,20);(9,12,13,11)(10,15,17,18)(14,19,21,20)(16,23,24,22)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,d^c=di,g^c=ghi,f^d=fh,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72278236, #72278265, #72281012, … |
| maximal normal subgroup of | #72278236, #72278265, #72281012, … |
| maximal quotient of | #72278765 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419231);