#119788

order	512
nilpotency class	3
isoclinism	isoclinic class of #119785
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

576 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26984);