#482366

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1061 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)(3,5)(4,6)(7,8)(9,10)(11,21,24,36)(12,25)(13,19,14,18)(15,20)(16,23)(17,29)(22,38,26,37)(27,34,28,35)(30,32)(31,40)(33,39);(2,5)(3,8)(6,7)(9,11)(10,16)(12,26)(13,23)(14,30)(15,24)(17,33)(18,21)(19,36)(20,32)(22,29)(25,31)(27,40)(28,39)(34,38)(35,37);(2,6)(5,7)(9,12,15,29)(10,17,20,25)(11,22,24,26)(13,27,14,28)(16,31,32,33)(18,34,19,35)(21,37,36,38)(23,39,30,40);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,23,24,30)(12,27,29,28)(16,21,32,36)(17,34,25,35)(22,39,26,40)(31,37,33,38)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=h,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, 389562);