#135492

order	512
nilpotency class	4
isoclinism	isoclinic class of #135471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1819
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

511 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=dh,e^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fg,f^c=fghi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42688);