#142962

order	512
nilpotency class	4
isoclinism	isoclinic class of #142936
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	#3459	C₂²
LCS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

588 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=g,e^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fhi,c^b=cf,d^b=dg,f^b=fgi,h^b=hi,d^c=dh,e^c=eg,f^c=fgi,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50158);