#493274

order	512
nilpotency class	2
isoclinism	isoclinic class of #180869
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

(1,2)(3,7)(4,6)(5,8)(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);(1,3)(2,5)(4,8)(6,7)(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);(3,8)(5,7);(2,6)(5,7)(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,e^2=i,f^2=h,h^2=i,b^a=be,d^a=deh,g^a=gi,c^b=ceh,d^b=di,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239455, #72239463, #72239467, …
maximal normal subgroup of	#72239455, #72239463, #72239467, …
maximal quotient of	#72343804

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400470);