#137326

order	512
nilpotency class	4
isoclinism	isoclinic class of #12096
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

565 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=gi,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,e^a=egh,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=eg,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44522);