#474315

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

552 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)(9,10,15,20,16,21,14,19)(11,22,17,12,25,27,24,28)(13,29,31,37,33,18,32,36)(23,38,34,26,35,30,39,40);(9,11,15,24,16,25,14,17)(10,12,20,22,21,28,19,27)(13,23,31,39,33,35,32,34)(18,26,36,38,29,40,37,30);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,20)(13,30)(14,27)(15,22)(16,28)(18,35)(19,25)(21,24)(23,29)(26,31)(32,40)(33,38)(34,36)(37,39);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,23)(12,26)(14,32)(15,31)(16,33)(17,34)(19,37)(20,36)(21,29)(22,38)(24,39)(25,35)(27,30)(28,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=egh,c^2=ef,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,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, 381511);