#139806

order	512
nilpotency class	4
isoclinism	isoclinic class of #139806
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9497
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#23904	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

718 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,11,15,28)(4,16,12,9)(5,19,21,8)(6,10)(7,13)(14,30)(17,24)(18,26)(20,27)(22,25)(23,29,32,31);(1,3,12,13,6,15,4,17)(2,7,16,28,10,24,9,11)(5,20,22,31,21,14,18,29)(8,23,25,30,19,32,26,27);(2,8,10,19)(5,21)(7,23,24,32)(9,26,16,25)(11,27,28,30)(14,20)(18,22)(29,31)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^d=fi,f^e=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, 47002);