#141338

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

292 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9,6,10,4,7)(3,11,14,16,12,15,13,8)(17,18,22,27,37,40,35,26,21,25,36,44,23,28,20,34)(19,29,32,24,39,42,31,46,30,47,48,41,33,45,38,43);(2,7,10,9)(3,12)(8,11,16,15)(13,14)(17,19,23,39,21,30,37,33)(18,24,40,43,25,41,28,46)(20,32,36,31,35,48,22,38)(26,45,44,47,34,42,27,29);(1,3,4,13,6,12,5,14)(2,8,9,11,10,16,7,15)(18,25)(19,30)(26,34)(27,44)(28,40)(31,38)(32,48)(33,39)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=ei,d^2=ghi,e^2=gi,f^2=gh,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.

Relationships

maximal subgroup of	#72287396, #72287440, #72287441
maximal normal subgroup of	#72287396, #72287440, #72287441

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48534);