#339889

order	512
nilpotency class	3
isoclinism	isoclinic class of #339847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

647 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 647 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247085);