#144054

order	512
nilpotency class	4
isoclinism	isoclinic class of #143949
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

404 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 84 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=gi,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,c^b=cf,d^b=dg,f^b=fgi,g^b=gi,d^c=dgi,e^c=eh,f^c=fgi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51250);