#136054

order	512
nilpotency class	4
isoclinism	isoclinic class of #12148
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	#1867
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

365 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,5)(4,7)(6,8)(9,10,13,18,14,19,26,34)(11,20,22,35,23,28,37,39)(12,17,25,33,24,16,31,32)(15,27,30,36,29,38,40,21);(2,5)(3,8)(6,7)(9,11,13,22,14,23,26,37)(10,15,18,30,19,29,34,40)(12,21,25,27,24,36,31,38)(16,28,32,39,17,20,33,35);(9,12,14,24)(10,16)(11,21,23,36)(13,25,26,31)(15,28)(17,19)(18,32)(20,29)(22,27,37,38)(30,39)(33,34)(35,40)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,d^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,d^b=dg,e^b=ehi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43250);