#474263

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

632 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efgh,c^2=e,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381459);