#244169

order	512
nilpotency class	3
isoclinism	isoclinic class of #232789
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

654 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

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

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, 151365);