#474662

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

701 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:

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

Polycyclic presentation:

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

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, 381858);