#146367

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×#97
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1162 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)(9,10,13,18,14,20,28,37,15,19,29,36,12,17,26,34)(11,21,24,38,25,16,30,32,22,33,27,40,23,31,39,35);(2,4)(9,11)(10,16,19,31)(12,23)(13,27)(14,25)(15,22)(17,21,20,33)(18,35,36,32)(24,29)(26,30)(28,39)(34,40,37,38);(6,8)(11,22)(16,31)(21,33)(23,25)(24,27)(30,39)(32,35)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287108, #72287113, #72287187
maximal normal subgroup of	#72287108, #72287113, #72287187
maximal quotient of	#72287342

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53563);