#142060

order	512
nilpotency class	4
isoclinism	isoclinic class of #142050
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1993
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 349 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=defg,c^2=eg,d^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,f^c=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, 49256);