#135401

order	512
nilpotency class	4
isoclinism	isoclinic class of #124865
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1525
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#8721
prime radical	C₂⁴	#101
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄² ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 295 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=d,d^2=i,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=eg,h^a=hi,f^b=fi,e^c=eh,f^c=fghi,g^c=gi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42597);