#136572

order	512
nilpotency class	4
isoclinism	isoclinic class of #136514
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1864
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#466	Q₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

258 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,d^2=g,e^2=h,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dgi,e^b=egi,d^c=dgi,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43768);