#205941

order	512
nilpotency class	3
isoclinism	isoclinic class of #205765
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

987 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113137);