#135549

order	512
nilpotency class	4
isoclinism	isoclinic class of #135518
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1819
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 642 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)(3,13,17,23)(4,12)(5,10,18,8)(6,11)(7,15,22,28)(9,14)(16,27,32,21)(19,24,29,25)(20,30,26,31);(1,3,14,28)(2,7,9,13)(4,15,6,17)(5,16,29,32)(8,20,24,26)(10,21,25,27)(11,22,12,23)(18,30,19,31);(2,8,9,24)(5,19)(7,20,13,26)(10,11,25,12)(16,31)(18,29)(21,22,27,23)(30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=dh,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=eg,f^c=fhi,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, 42745);