#138899

order	512
nilpotency class	4
isoclinism	isoclinic class of #138768
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#554 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 506 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) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=ghi,d^2=gh,e^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fh,g^b=gi,h^b=hi,d^c=dgh,e^c=egh,f^c=fgh,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, 46095);