#139453

order	512
nilpotency class	4
isoclinism	isoclinic class of #139432
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

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

normal subgroups only

This group has 556 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=i,d^2=gh,f^2=i,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, 46649);