#142971

order	512
nilpotency class	4
isoclinism	isoclinic class of #142969
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	#3490	C₂²
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 615 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50167);