#142979

order	512
nilpotency class	4
isoclinism	isoclinic class of #142966
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×C₄⋊C₄	C₂³
prime radical	#3492	C₂²
LCS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 591 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fg,h^b=hi,d^c=dh,f^c=fg,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, 50175);