#244164

order	512
nilpotency class	3
isoclinism	isoclinic class of #232755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2397
prime radical	#3150	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

659 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 659 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=h,c^2=h,d^2=efh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi

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, 151360);