#139754

order	512
nilpotency class	4
isoclinism	isoclinic class of #139751
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9495
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#23913	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 706 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=gi,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, 46950);