#139430

order	512
nilpotency class	4
isoclinism	isoclinic class of #139428
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9428
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₄×M₁₆	C₂³
prime radical	#19721	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

349 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 349 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,d^2=gh,e^2=g,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=eg,f^c=fgh,h^c=hi,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, 46626);