#142059

order	512
nilpotency class	4
isoclinism	isoclinic class of #125422
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
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×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

346 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 82 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)(4,5)(6,8)(9,10)(11,21,18,31)(12,17)(13,29,30,24)(14,19)(15,20)(16,23)(22,36,33,40)(25,26,38,39)(27,34)(28,35)(32,37);(2,5)(6,7)(9,11,21,10,16,24,30,19,15,18,31,20,14,29,13,23)(12,25,36,34,32,33,39,35,28,38,40,37,27,22,26,17);(1,3,4,8)(2,6,5,7)(9,12,16,32,15,28,14,27)(10,17,23,37,20,35,19,34)(11,22,29,38,18,33,24,25)(13,26,21,36,30,39,31,40)

Polycyclic presentation:

2^9|a^2=f,b^2=defg,c^2=egi,d^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,h^b=hi,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, 49255);