#495475

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

548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=f,e^2=h,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	#72313379, #72313451, #72313474, …
maximal normal subgroup of	#72313379, #72313451, #72313474, …
maximal quotient of	#72343685, #72343775

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402671);