#409656

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,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, 316852);