#445798

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

828 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=fg,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^d=ei,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, 352994);