#139752

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	#23915	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 698 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)(3,5,13,19)(4,14)(6,9,22,26)(7,10)(8,16,24,21)(11,27)(12,29)(15,30,31,28)(17,25)(18,20)(23,32);(1,3,5,10,7,13,19,2)(4,11,15,24,17,23,31,8)(6,20,29,26,22,12,18,9)(14,16,30,32,25,21,28,27);(1,4)(2,8)(3,11)(5,15)(6,21)(7,17)(9,25)(10,24)(12,30)(13,23)(14,26)(16,22)(18,32)(19,31)(20,28)(27,29)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,e^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, 46948);