#6383749

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator, socle	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

(2,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,24,21,19,20,23);(1,2,5,6)(3,7,4,8)(9,11,14,10)(12,16,13,15)(17,19,22,23,21,18,20,24);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,15,11,16)(18,19)(23,24);(1,3,5,4)(2,8,6,7)(9,13,14,12)(10,15,11,16);(1,4,5,3)(2,8,6,7)(17,20,21,22)(18,23,19,24)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=f,d^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,c^b=ci,d^b=df

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6290945);