#244129

order	512
nilpotency class	3
isoclinism	isoclinic class of #232766
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	#10032	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

692 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^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,e^b=eh,f^b=fh,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, 151325);