#415530

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

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

580 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:

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

Polycyclic presentation:

2^9|a^2=efgh,b^2=ef,c^2=e,d^2=h,e^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,d^b=di,f^c=fi,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, 322726);