#238120

order	512
nilpotency class	3
isoclinism	isoclinic class of #234066
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

918 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 202 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=dh,f^c=fh,g^c=gi

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, 145316);