#495478

order	512
nilpotency class	4
isoclinism	isoclinic class of #483425
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

558 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)(3,6)(4,5)(7,8)(9,10,13,19,15,20,14,29)(11,17,23,33,21,31,24,39)(12,25,27,34,28,16,26,18)(22,36,38,40,35,30,37,32);(10,16,20,25)(12,26)(13,14)(17,30,31,36)(18,19,34,29)(22,37)(23,24)(27,28)(32,33,40,39)(35,38);(2,5)(6,8)(9,11)(10,17)(12,22)(13,23)(14,24)(15,21)(16,30)(18,32)(19,33)(20,31)(25,36)(26,37)(27,38)(28,35)(29,39)(34,40);(1,3,4,7)(2,6,5,8)(11,21)(17,31)(22,35)(23,24)(30,36)(32,40)(33,39)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=f,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237237, #72239543, #72239558, …
maximal normal subgroup of	#72237237, #72239543, #72239558, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402674);