#326942

order	512
nilpotency class	3
isoclinism	isoclinic class of #326925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

816 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=ghi,d^c=de,f^c=fh,g^c=ghi

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, 234138);