#119124

order	512
nilpotency class	3
isoclinism	isoclinic class of #119122
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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

553 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)(9,10,14,20,15,18,13,19)(11,21,24,32,25,16,23,33)(12,26,28,35,30,36,29,17)(22,38,39,31,40,34,27,37);(1,3,6,2)(4,5,8,7)(9,11,15,25)(10,16,18,21)(12,27,30,39)(13,24,14,23)(17,34,35,38)(19,33,20,32)(22,28,40,29)(26,31,36,37);(1,4,5,2,6,8,7,3)(9,12)(10,17,18,35)(11,22)(13,28)(14,29)(15,30)(16,31,21,37)(19,26,20,36)(23,39)(24,27)(25,40)(32,34,33,38)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=hi,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=ei,f^c=fhi,g^c=gh

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, 26320);