#498236

order	512
nilpotency class	4
isoclinism	isoclinic class of #484039
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

543 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)(6,8)(9,10,14,18,15,20,12,26)(11,21,25,33,22,37,23,16)(13,28,30,35,31,17,27,19)(24,36,39,40,29,34,38,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16,20,33)(12,23)(13,29)(14,25)(15,22)(17,32,28,40)(18,21,26,37)(19,36,35,34)(24,31)(27,39)(30,38);(1,3,4,7)(2,6,5,8)(10,17)(12,14)(13,30)(16,32)(18,35)(19,26)(20,28)(21,34)(23,25)(24,39)(27,31)(29,38)(33,40)(36,37);(2,5)(6,8)(11,22)(16,33)(21,37)(23,25)(24,29)(32,40)(34,36)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309445, #72309603, #72310198, …
maximal normal subgroup of	#72309445, #72309603, #72310198, …
maximal quotient of	#72343498, #72343592

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405432);