#443339

order	512
nilpotency class	3
isoclinism	isoclinic class of #423425
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efg,c^2=efi,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 350535);