#205948

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1000 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,d^2=i,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, 113144);