#135402

order	512
nilpotency class	4
isoclinism	isoclinic class of #124987
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1536
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 ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 290 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=d,d^2=i,e^2=hi,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,h^a=hi,e^c=eh,f^c=fgh,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, 42598);