#140241

order	512
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1677
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

249 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2,4,3)(5,6)(7,17)(8,23,27,40)(9,28)(10,32)(11,15)(12,20)(13,26,39,60)(14,31)(16,30)(18,37)(19,48,52,67)(21,45)(22,57)(24,33)(25,38,29,41)(34,63,66,68)(35,46)(36,49)(42,51,64,53)(43,56)(44,55)(47,59,61,62)(50,58,54,65);(1,3,4,2)(5,7,18,46,10,22,9,20,11,21,31,56,30,55,24,49)(6,12,33,57,16,35,14,17,15,36,28,44,32,43,37,45)(8,19,47,63,26,53,29,54,27,52,61,68,60,51,25,50)(13,34,62,67,40,65,41,42,39,66,59,48,23,58,38,64);(5,8,24,25,30,60,31,61,11,27,9,29,10,26,18,47)(6,13,37,38,32,23,28,59,15,39,14,41,16,40,33,62)(7,19,49,50,55,51,56,68,21,52,20,54,22,53,46,63)(12,34,45,64,43,58,44,48,36,66,17,42,35,65,57,67)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=dh,d^2=gi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=ei,d^c=di,e^c=eghi,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, 47437);