#482385

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
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	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂×#532	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1009 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)(3,5)(4,6)(7,8)(9,10,13,18,14,17,27,32)(11,19,22,33,20,15,28,30)(12,23,24,34,25,16,26,31)(21,29,37,39,36,35,38,40);(2,5)(3,8)(6,7)(9,11)(10,15)(12,21)(13,22)(14,20)(16,29)(17,19)(18,33)(23,35)(24,37)(25,36)(26,38)(27,28)(30,32)(31,40)(34,39);(2,6)(5,7)(11,20)(15,19)(21,36)(22,28)(29,35)(30,33)(37,38)(39,40);(2,6)(5,7)(9,12,14,25)(10,16,17,23)(11,21,20,36)(13,26,27,24)(15,29,19,35)(18,34,32,31)(22,38,28,37)(30,40,33,39)

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,f^d=fi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389581);