#475462

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

700 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)(11,16)(12,24)(13,20)(14,18)(15,19)(17,25)(21,36)(22,30)(23,32)(26,34)(27,31)(28,35)(29,37)(33,38)(39,40);(2,5)(6,8)(9,11,14,22,15,23,13,27)(10,16,18,30,19,32,20,31)(12,21,25,38,26,39,28,37)(17,33,34,40,35,29,24,36);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,28)(14,25)(15,26)(16,29)(18,34)(19,35)(20,24)(22,38)(23,39)(27,37)(30,36)(31,40)(32,33);(3,7)(6,8)(10,18,19,20)(16,30,32,31)(17,34,35,24)(29,36,33,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=e,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,f^c=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, 382658);