#495322

order	512
nilpotency class	4
isoclinism	isoclinic class of #483839
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

538 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)(11,19,21,34)(13,25)(14,24)(16,29,27,30)(17,31)(20,36,37,33)(22,39,38,35)(23,40)(26,32);(2,5)(6,8)(9,10)(11,20,21,37)(12,15)(13,26)(14,18)(16,22,27,38)(17,23)(19,33,34,36)(24,28)(25,31)(29,35,30,39)(32,40);(9,11,13,19,14,22,25,35,12,21,23,34,24,38,40,39)(10,16,17,30,18,20,32,36,15,27,26,29,28,37,31,33);(1,3,4,7)(2,6,5,8)(10,15)(16,27)(17,26)(18,28)(20,37)(29,30)(31,32)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313240, #72313273, #72313285, …
maximal normal subgroup of	#72313240, #72313273, #72313285, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402518);