#481550

order	512
nilpotency class	4
isoclinism	isoclinic class of #481477
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

887 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)(9,10,14,19,15,18,27,33)(11,20,22,34,12,16,24,29)(13,17,25,32,26,31,37,40)(21,30,35,39,23,28,36,38);(1,2)(3,6)(4,8)(5,7)(9,11,15,12)(10,16,18,20)(13,23,26,21)(14,22,27,24)(17,30,31,28)(19,34,33,29)(25,36,37,35)(32,38,40,39);(1,3,5,4)(2,6,7,8)(9,12,15,11)(10,16,18,20)(13,21,26,23)(14,24,27,22)(17,30,31,28)(19,29,33,34)(25,35,37,36)(32,39,40,38);(9,13,15,26)(10,17,18,31)(11,21,12,23)(14,25,27,37)(16,28,20,30)(19,32,33,40)(22,35,24,36)(29,38,34,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388746);