#409769

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

559 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:

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

Polycyclic presentation:

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

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, 316965);