#139475

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 573 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)(3,13,15,28)(4,9,18,21)(5,14,24,10)(6,12)(7,26)(8,11)(16,30,17,22)(19,32,20,27)(23,31,25,29);(1,3,5,21,7,15,25,27)(2,4,14,22,26,19,29,30)(6,17,23,9,8,16,24,32)(10,28,12,18,31,13,11,20);(1,4,8,20)(2,9,11,27)(3,14,16,10)(5,22,24,28)(6,18,7,19)(12,21,26,32)(13,25,30,23)(15,29,17,31)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=i,d^2=ghi,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=fhi,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, 46671);