#339891

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
prime radical	#10280	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 619 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)(3,10)(4,13)(5,9)(6,15)(7,12)(8,11)(14,16)(17,18,23,25)(19,26,28,30)(20,27,31,21)(22,29,32,24);(1,3,7,13)(2,9,12,16)(4,15,10,11)(5,6,14,8)(17,19)(18,20)(21,32)(22,27)(23,28)(24,30)(25,31)(26,29);(1,4,7,10)(2,3,12,13)(5,15,14,11)(6,9,8,16)(17,20,23,31)(18,19,25,28)(21,29,27,24)(22,26,32,30);(1,5,6,13,7,14,8,3)(2,10,11,16,12,4,15,9)(17,21,32,19,23,27,22,28)(18,20,29,30,25,31,24,26)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=ei,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, 247087);