#351619

order	512
nilpotency class	3
isoclinism	isoclinic class of #347788
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

545 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,5)(4,6)(7,8)(10,14)(11,17)(12,20)(15,21)(16,23)(18,22)(25,26,28,31)(27,30,34,38)(29,32,36,33)(35,37,40,39);(1,3,4,7)(2,5,6,8)(9,10,12,14,13,15,20,21)(11,18,19,23,17,16,24,22)(25,27)(26,30)(28,34)(29,35)(31,38)(32,37)(33,39)(36,40);(2,6)(5,8)(10,15)(11,17)(14,21)(19,24)(26,31)(27,34)(32,33)(35,40);(9,11,12,19,13,17,20,24)(10,16,14,22,15,18,21,23)(26,32)(29,36)(30,37)(31,33)(35,40)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=efi,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=chi,d^b=dh,d^c=dh,g^d=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258815);