#136811

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	#3002	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 516 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)(3,7,8,5)(4,6)(9,10,14,18,13,19,28,36,15,20,30,37,29,17,33,35)(11,12,22,25,21,26,32,39,23,27,34,40,38,24,16,31);(1,2)(3,5)(4,6)(7,8)(9,11,14,22,13,21,28,32,15,23,30,34,29,38,33,16)(10,12,18,25,19,26,36,39,20,27,37,40,17,24,35,31);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,18)(13,26)(14,25)(15,27)(17,34)(19,22)(20,32)(21,36)(23,37)(24,29)(28,39)(30,40)(31,33)(35,38)

Polycyclic presentation:

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

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, 44007);