#244155

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

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

normal subgroups only

This group has 678 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^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, 151351);