Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499803 |
| 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 | #2404 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
980 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)(3,8)(4,6)(5,7)(10,12)(11,17)(15,22)(16,20)(18,23)(19,24);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,13)(14,20)(16,21)(17,23)(18,19)(22,24);(1,4,5,3)(2,6,7,8)(9,11,13,19)(10,15,12,18)(14,17,21,24)(16,23,20,22);(2,7)(6,8)(9,12,13,10)(11,18,19,15)(14,20,21,16)(17,22,24,23)Polycyclic presentation:
2^9|b^2=ei,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72311968, #72312010, #72312269, … |
| maximal normal subgroup of | #72311968, #72312010, #72312269, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407604);