#136760

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	#447	C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

342 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=egi,d^b=dg,e^b=ei,f^b=fg,d^c=di,e^c=eg,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43956);