#139921

order	512
nilpotency class	4
isoclinism	isoclinic class of #139718
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 574 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=i,d^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dgi,e^b=eh,f^b=fgi,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, 47117);