#411145

order	512
nilpotency class	3
isoclinism	isoclinic class of #401009
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

734 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)(3,8)(4,7)(5,6)(9,10,13,31,15,20,14,19)(11,21,23,16,25,33,24,18)(12,26,28,36,30,17,29,37)(22,34,27,32,40,38,39,35);(2,6)(7,8)(9,11)(10,16,20,18)(12,27)(13,23)(14,24)(15,25)(17,34,26,38)(19,21,31,33)(22,29)(28,40)(30,39)(32,36,35,37);(1,3,5,4)(2,7,6,8)(9,12,13,29,15,30,14,28)(10,17,31,36,20,26,19,37)(11,22,24,27,25,40,23,39)(16,32,21,38,18,35,33,34);(1,4,5,3)(2,7,6,8)(9,11)(10,18,20,16)(12,22)(13,24)(14,23)(15,25)(17,35,26,32)(19,21,31,33)(27,29)(28,39)(30,40)(34,37,38,36)

Polycyclic presentation:

2^9|a^2=ef,b^2=gh,c^2=fg,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318341);