#494102

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

622 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)(9,10,14,20,13,19,28,36,15,21,30,17,29,18,12,27)(11,22,26,38,25,16,31,35,24,34,40,37,39,33,23,32);(1,3,4,7)(2,5,6,8)(9,11,15,24)(10,16)(12,23,28,31)(13,25,29,39)(14,26,30,40)(17,32)(18,22)(19,34)(20,35)(21,33)(27,38)(36,37);(2,6)(5,8)(9,12,13,14,15,28,29,30)(10,17,18,36,21,20,19,27)(11,23,25,26,24,31,39,40)(16,32,22,37,33,35,34,38);(11,24)(16,33)(22,34)(23,31)(25,39)(26,40)(32,35)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239996, #72240008, #72264495, …
maximal normal subgroup of	#72239996, #72240008, #72264495, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401298);