#201680

order	512
nilpotency class	3
isoclinism	isoclinic class of #201678
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2040 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)(3,7,8,5)(4,6)(9,10)(11,15,21,28)(12,16)(13,17)(14,18)(19,26,34,38)(20,27,36,40)(22,29)(23,30)(24,31)(25,32)(33,37,35,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15,18,28)(12,19)(13,20)(14,21)(16,26,30,38)(17,27,32,40)(22,33)(23,34)(24,35)(25,36)(29,37,31,39);(1,3)(2,5)(4,8)(6,7)(9,12,14,23)(10,16,18,30)(11,19,21,34)(13,24,25,22)(15,26,28,38)(17,31,32,29)(20,35,36,33)(27,39,40,37);(9,13,14,25)(10,17,18,32)(11,20,21,36)(12,22,23,24)(15,27,28,40)(16,29,30,31)(19,33,34,35)(26,37,38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,g^c=gi

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, 108876);