#135683

order	512
nilpotency class	4
isoclinism	isoclinic class of #135683
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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

226 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 114 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)(3,12)(4,15)(5,16)(6,7)(10,11)(13,14)(18,24,27,26)(19,29,28,30)(20,25)(21,32)(22,31);(1,2,5,10,6,11,4,9,8,13,15,3,7,12,16,14)(17,18,21,26,22,28,32,30,23,27,20,24,31,19,25,29);(1,3,4,2,7,9,5,12,8,10,16,13,6,14,15,11)(17,19,20,30,22,18,25,24,23,28,21,29,31,27,32,26)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42879);