#497055

order	512
nilpotency class	4
isoclinism	isoclinic class of #483993
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×#493	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

583 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264915, #72264975, #72265360, …
maximal normal subgroup of	#72264915, #72264975, #72265360, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404251);