#494127

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

560 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)(3,5)(4,6)(7,8)(9,10,12,18,13,19,26,35,14,17,25,33,27,16,31,32)(11,20,22,34,23,28,37,39,21,15,24,29,36,30,38,40);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,24)(13,23)(14,21)(16,28)(17,20)(18,34)(19,30)(22,25)(26,38)(27,36)(29,33)(31,37)(32,40)(35,39);(2,6)(5,8)(11,21)(15,20)(22,24)(23,36)(28,30)(29,34)(37,38)(39,40);(10,16,17,19)(12,25)(15,28,20,30)(18,35,33,32)(22,24)(26,31)(29,40,34,39)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=ci,g^b=gi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239990, #72239996, #72265493, …
maximal normal subgroup of	#72239990, #72239996, #72265493, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401323);