#189651

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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×#380
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

914 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:

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

Polycyclic presentation:

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

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, 96847);