#413577

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	#2537
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

754 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)(11,23)(12,28)(13,33)(14,21)(15,20)(16,22)(17,25)(18,31)(19,34)(24,40)(26,36)(27,35)(29,39)(30,38)(32,37);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,29)(13,32)(14,26)(15,25)(16,27)(18,33)(19,37)(20,35)(21,23)(22,36)(24,31)(28,40)(30,34)(38,39);(1,4,5,3)(2,6,7,8)(9,12)(10,18)(11,24)(13,26)(14,31)(15,30)(16,32)(17,19)(20,28)(21,37)(22,38)(23,39)(25,29)(27,34)(33,35)(36,40);(9,13,14,34,16,29,15,24)(10,19,20,33,22,40,21,39)(11,12,26,31,27,32,25,30)(17,28,35,38,36,37,23,18)

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=ef,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,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, 320773);