#495621

order	512
nilpotency class	4
isoclinism	isoclinic class of #480476
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

939 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402817);