Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499831 |
| 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×#453 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(10,12)(14,22)(15,18)(17,24);(1,3,5,4)(2,6,8,7)(9,10)(11,17)(12,13)(14,21)(15,20)(16,22)(18,23)(19,24);(9,11)(10,15)(12,18)(13,19)(14,23)(16,24)(17,21)(20,22);(1,4,5,3)(2,7,8,6)(9,12,13,10)(11,18,19,15)(14,21,22,16)(17,20,24,23)Polycyclic presentation:
2^9|a^2=ei,b^2=ei,d^2=e,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 | #72285788, #72285873, #72285978, … |
| maximal normal subgroup of | #72285788, #72285873, #72285978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408163);