#496695

order	512
nilpotency class	4
isoclinism	isoclinic class of #483439
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 ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

525 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)(3,7)(4,6)(5,8)(9,10)(11,19,20,31)(12,15)(13,25)(14,27)(16,21,28,33)(17,26)(18,24)(22,36,34,32)(23,37)(29,38)(30,40,39,35);(1,3,4,8)(2,5,6,7)(10,15)(11,20)(17,29)(18,27)(21,33)(22,34)(25,37)(35,40);(3,8)(5,7)(9,11,13,21,14,22,26,35,12,20,23,33,24,34,38,40)(10,16,17,19,18,30,25,32,15,28,29,31,27,39,37,36);(2,6)(5,7)(10,15)(16,28)(17,29)(18,27)(19,31)(25,37)(30,39)(32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=f,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313363, #72313467, #72313474, …
maximal normal subgroup of	#72313363, #72313467, #72313474, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403891);