#136816

order	512
nilpotency class	3
isoclinism	isoclinic class of #9519
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
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×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

505 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 88 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)(9,10,15,20,14,19,33,37,16,21,35,39,13,32,23,24)(11,22,26,40,28,12,29,31,27,30,18,38,25,17,36,34);(1,2)(3,5)(4,6)(7,8)(9,11,23,36,13,25,35,18,16,27,33,29,14,28,15,26)(10,17,37,40,32,30,20,34,21,12,24,38,19,22,39,31);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,24)(13,22)(14,30)(15,34)(16,17)(19,36)(20,28)(21,26)(23,38)(25,39)(27,37)(29,32)(31,35)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44012);