#415558

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
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	#2563
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

692 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)(11,16,24,35)(12,27,23,17)(13,30)(14,19)(15,20)(18,32)(21,33)(22,39)(25,36,26,40)(28,34,29,38)(31,37);(1,3,5,4)(2,6,7,8)(9,11,14,24)(10,16,19,35)(12,13,23,31)(15,26,33,25)(17,37,27,30)(18,34,39,38)(20,40,21,36)(22,29,32,28);(1,4,5,3)(2,6,7,8)(9,12,14,23)(10,17,19,27)(11,22,24,32)(13,25,31,26)(15,29,33,28)(16,18,35,39)(20,34,21,38)(30,40,37,36);(9,13,15,32,14,31,33,22)(10,18,21,30,19,39,20,37)(11,23,26,28,24,12,25,29)(16,34,36,17,35,38,40,27)

Polycyclic presentation:

2^9|a^2=egh,b^2=ef,c^2=ef,d^2=h,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,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322754);