#161404

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=g,e^2=h,f^2=i,b^a=be,c^a=cg,e^a=eg,c^b=ci,e^b=eh,f^b=fgh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68600);