#139847

order	512
nilpotency class	4
isoclinism	isoclinic class of #139712
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂⁵	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

715 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,14,19,26)(4,21,24,9)(5,25,27,16)(6,11)(7,13)(8,17,29,30)(10,28,32,15)(12,22)(18,31,20,23);(1,3,15,17,7,20,28,21)(2,8,25,23,13,4,16,14)(5,9,6,18,27,30,22,19)(10,31,11,29,32,26,12,24);(1,4,22,29)(2,9,12,17)(3,16,19,32)(5,26,15,14)(6,24,7,8)(10,20,25,18)(11,21,13,30)(23,27,31,28)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,f^c=fh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47043);