#151101

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₄	C₂²×C₃₂
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	D₄×C₁₆
prime radical	C₂×D₈	C₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,6)(3,5,8,7)(9,10,12,17,13,18,23,30,14,19,24,31,25,32,36,39,15,11,16,20,26,21,28,33,27,22,29,34,37,35,38,40);(2,5)(3,8)(6,7);(2,6)(5,7)(9,11,12,20,13,21,23,33,14,22,24,34,25,35,36,40,15,10,16,17,26,18,28,30,27,19,29,31,37,32,38,39)

Polycyclic presentation:

2^9|a^2=e,c^2=ef,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=cf,d^a=df,d^b=df

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72230710
maximal normal subgroup of	#72230710
maximal quotient of	#72230710, #72230752

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58297);