#504473

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13319	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

590 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)(9,10,17,35,15,23,27,18,14,22,13,21,16,24,33,37)(11,25,32,39,34,40,12,30,29,19,28,38,26,20,31,36);(10,18)(11,26)(13,27)(15,16)(17,33)(19,38)(20,30)(21,24)(22,37)(23,35)(25,39)(28,32)(29,34)(36,40);(2,5)(6,8)(9,11)(10,19)(12,27)(13,28)(14,29)(15,34)(16,26)(17,32)(18,36)(20,23)(21,39)(22,25)(24,40)(30,37)(31,33)(35,38);(1,3,4,7)(2,6,5,8)(9,12)(10,20)(11,27)(13,26)(14,31)(15,32)(16,28)(17,34)(18,30)(19,23)(21,39)(22,40)(24,25)(29,33)(35,38)(36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411669);