#331320

order	512
nilpotency class	3
isoclinism	isoclinic class of #331304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

711 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 711 conjugacy classes of subgroups — too many to draw. Only its 162 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)(9,10,13,16,14,17,12,21)(11,18,19,24,20,23,22,15)(25,26)(27,35)(28,33)(29,32)(30,34)(31,38)(36,39)(37,40);(2,5)(6,8)(25,27,30,37,29,38,28,36)(26,31,34,39,32,35,33,40);(1,3)(2,6)(4,7)(5,8)(9,11,14,20)(10,15,17,24)(12,19,13,22)(16,23,21,18)(26,32)(27,36)(28,30)(31,39)(35,40)(37,38);(3,7)(6,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(25,28,29,30)(26,33,32,34)(27,37,38,36)(31,39,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238516);