#497051

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	#3147	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

547 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,d^2=ei,e^2=i,f^2=h,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	#72343219, #72343238, #72343250
maximal normal subgroup of	#72343219, #72343238, #72343250
maximal quotient of	#72343320, #72343394

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404247);