#119691

order	512
nilpotency class	3
isoclinism	isoclinic class of #119683
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

660 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,f^b=fh,g^b=gh,e^c=ehi,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26887);