#497708

order	512
nilpotency class	4
isoclinism	isoclinic class of #484095
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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

662 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)(3,5)(4,6)(7,8)(9,10,15,21,16,22,12,18)(11,23,24,38,27,30,34,40)(13,19,20,33,31,29,28,14)(17,32,35,26,25,39,36,37);(1,3,4,7)(2,5,6,8)(9,11,16,27)(10,17,22,25)(12,24,15,34)(13,30,31,23)(14,32,33,39)(18,35,21,36)(19,37,29,26)(20,38,28,40);(2,6)(5,8)(9,12,16,15)(10,18,22,21)(11,24,27,34)(13,28,31,20)(14,29,33,19)(17,35,25,36)(23,38,30,40)(26,39,37,32);(9,13)(10,19)(11,25)(12,28)(14,18)(15,20)(16,31)(17,27)(21,33)(22,29)(23,39)(24,36)(26,40)(30,32)(34,35)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,e^d=ehi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310208, #72310222, #72313244, …
maximal normal subgroup of	#72310208, #72310222, #72313244, …
maximal quotient of	#72343329, #72343409

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404904);