#244156

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	#10032	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 681 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=ef,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, 151352);