#142995

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

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

normal subgroups only

This group has 569 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,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, 50191);