#108158

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂×#121	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

772 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:

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

Polycyclic presentation:

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

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, 15354);