#142891

order	512
nilpotency class	4
isoclinism	isoclinic class of #142220
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

621 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 621 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,c^b=cf,d^b=dg,f^b=fg,h^b=hi,d^c=dh,f^c=fg,f^d=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, 50087);