#244126

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	#3150	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 670 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:

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

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,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, 151322);