#139449

order	512
nilpotency class	4
isoclinism	isoclinic class of #139432
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 662 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)(3,13,18,30)(4,16,20,28)(5,10,14,15)(6,12)(7,11)(8,26)(9,19,22,21)(17,31,27,23)(24,29,25,32);(1,3,14,28,8,27,24,16)(2,4,15,30,26,20,29,31)(5,22,6,18,25,9,7,17)(10,23,12,19,32,13,11,21);(1,4,7,21)(2,9,11,16)(3,15,17,10)(5,23,14,30)(6,19,8,20)(12,28,26,22)(13,24,31,25)(18,32,27,29)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=i,d^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gi,h^a=hi,d^b=dh,e^b=eghi,f^b=fh,g^b=gi,h^b=hi,d^c=dghi,f^c=fghi,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, 46645);