#135697

order	512
nilpotency class	4
isoclinism	isoclinic class of #135685
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1834
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#486
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

214 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42893);