#244150

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
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

687 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 687 conjugacy classes of subgroups — too many to draw. Only its 150 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)(9,10,14,19,15,20,12,17)(11,21,22,32,24,30,29,16)(13,18,27,35,28,36,25,33)(23,37,38,34,26,40,39,31);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,22)(13,26)(14,29)(15,24)(17,21)(18,34)(19,30)(20,32)(23,28)(25,39)(27,38)(31,36)(33,40)(35,37);(1,3,6,2)(4,7,5,8)(9,12,15,14)(10,17,20,19)(11,22,24,29)(13,27,28,25)(16,30,32,21)(18,35,36,33)(23,39,26,38)(31,37,34,40);(1,4,6,5)(2,7,3,8)(9,13,12,25,15,28,14,27)(10,18,17,33,20,36,19,35)(11,23,22,38,24,26,29,39)(16,31,21,37,32,34,30,40)

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=efh,e^2=h,f^2=i,b^a=be,c^a=cf,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151346);