#497988

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	C₂×#509	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

556 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)(4,8)(5,7)(9,10,14,20,13,17,15,29)(11,18,23,35,22,32,30,34)(12,24,25,33,27,16,28,19)(21,37,39,31,38,36,26,40);(1,3,5,4)(2,6,7,8)(10,16)(11,21)(12,25)(14,15)(17,24)(18,34)(19,29)(20,33)(22,38)(23,26)(27,28)(30,39)(32,35)(36,37);(1,4,5,3)(2,6,7,8)(10,17)(16,24)(18,32)(19,33)(20,29)(31,40)(34,35)(36,37);(9,11)(10,18)(12,26)(13,22)(14,23)(15,30)(16,31)(17,32)(19,36)(20,35)(21,25)(24,40)(27,39)(28,38)(29,34)(33,37)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=fi,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	#72309713, #72310202, #72310208, …
maximal normal subgroup of	#72309713, #72310202, #72310208, …
maximal quotient of	#72343501, #72343590

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405184);