#493146

order	512
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂²×D₈
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=ci,e^b=eh,f^b=fh,h^b=hi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320128, #72322680, #72323175
maximal normal subgroup of	#72320128, #72322680, #72323175
maximal quotient of	#72320205

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400342);