#141288

order	512
nilpotency class	4
isoclinism	isoclinic class of #141286
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1982
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#460
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

342 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 102 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)(3,11)(4,7)(5,14)(6,10)(8,12)(9,15)(13,16)(17,18,23,27,22,32,20,30,25,28,21,26,24,19,29,31);(1,3,5,4,6,12,15,13)(2,7,9,11,10,16,14,8)(17,19,25,32)(18,22,28,24)(20,30,29,31)(21,27,23,26);(1,4,5,12,6,13,15,3)(2,8,9,7,10,11,14,16)(17,20,24,23,25,29,22,21)(18,26,32,31,28,27,19,30)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=ei,d^2=ghi,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=cg,d^b=dh,f^b=fh,h^b=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, 48484);