#205115

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	#60673	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

986 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112311);