#437922

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1112 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345118);