#481654

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

919 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=cg,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fghi,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, 388850);