#139945

order	512
nilpotency class	4
isoclinism	isoclinic class of #139712
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	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

678 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^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=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, 47141);