#481777

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

912 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:

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

Polycyclic presentation:

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

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, 388973);