#139909

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, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

901 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 901 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=i,f^2=h,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, 47105);