#499564

order	512
nilpotency class	4
isoclinism	isoclinic class of #13398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#481
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

558 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,c^2=e,d^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=ghi,h^c=hi

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, 406760);