#331214

order	512
nilpotency class	3
isoclinism	isoclinic class of #331190
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

710 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 178 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)(9,10,14,18,13,17,12,16)(11,19,20,23,21,15,22,24);(2,5)(6,8)(9,11)(10,15)(12,22)(13,21)(14,20)(16,23)(17,19)(18,24)(25,26,28,31,29,32,37,40)(27,33,34,38,35,39,36,30);(1,3)(2,6)(4,7)(5,8)(10,16)(12,14)(15,23)(17,18)(19,24)(20,22)(25,27,29,35)(26,30,32,38)(28,36,37,34)(31,39,40,33);(3,7)(6,8)(9,12,13,14)(10,16,17,18)(11,20,21,22)(15,24,19,23)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,d^c=de,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, 238410);