#471044

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

628 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)(3,6)(4,5)(7,8)(9,10,16,19,15,20,14,30)(11,21,29,33,23,37,13,17)(12,24,31,36,27,18,26,35)(22,38,28,40,25,34,39,32);(9,11,14,13,15,23,16,29)(10,17,19,21,20,33,30,37)(12,25,31,39,27,22,26,28)(18,32,36,34,24,40,35,38);(2,5)(6,8)(9,12)(10,18)(11,22)(13,28)(14,31)(15,27)(16,26)(17,32)(19,36)(20,24)(21,34)(23,25)(29,39)(30,35)(33,40)(37,38);(1,3,4,7)(2,6,5,8)(9,13,14,23,15,29,16,11)(10,17,30,37,20,33,19,21)(12,25,26,28,27,22,31,39)(18,34,36,40,24,38,35,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378240);