#136251

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

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	C₄×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

346 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=h,d^2=g,e^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,d^c=dh,e^c=eg,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, 43447);