#142184

order	512
nilpotency class	4
isoclinism	isoclinic class of #125419
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1514
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 349 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=e,d^2=gh,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,d^c=di,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49380);