#494989

order	512
nilpotency class	4
isoclinism	isoclinic class of #483807
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×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

567 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)(10,16)(11,21,23,39)(12,14)(13,26)(17,32,33,36)(18,20)(22,40,24,28)(25,29)(30,35,34,38)(31,37);(2,5)(6,8)(9,10,14,20,15,18,12,16)(11,17,22,34,23,33,24,30)(13,27,26,31,29,19,25,37)(21,38,28,32,39,35,40,36);(9,11)(10,17)(12,24)(13,28)(14,22)(15,23)(16,30)(18,33)(19,36)(20,34)(21,25)(26,39)(27,32)(29,40)(31,35)(37,38);(1,3,4,7)(2,6,5,8)(10,18)(16,20)(17,33)(19,27)(30,34)(31,37)(32,36)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343225, #72343235, #72343260
maximal normal subgroup of	#72343225, #72343235, #72343260
maximal quotient of	#72343530, #72343560

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402185);