#118927

order	512
nilpotency class	3
isoclinism	isoclinic class of #118926
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

559 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,e^c=eh,f^c=fhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26123);