#146319

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₄	C₄²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₈×D₄
prime radical	C₂⁴	#131
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=i,e^2=h,g^2=h,h^2=i,b^a=bd,d^a=di,f^a=fi,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287066, #72287665
maximal normal subgroup of	#72287066, #72287665

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53515);