#121765

order	512
nilpotency class	3
isoclinism	isoclinic class of #112888
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28961);