Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,13)(12,20,22,16)(14,17)(15,19)(18,21,24,23);(2,6)(4,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,21)(22,24);(1,2)(3,4)(5,6)(7,8)(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,20)(19,24);(1,3,5,8)(2,7,6,4)(9,13,14,15)(10,11,17,19)(12,21,22,23)(16,18,20,24);(1,4,5,7)(2,3,6,8)Polycyclic presentation:
2^9|a^2=fi,d^2=hi,e^2=h,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524677);