#141219

order	512
nilpotency class	4
isoclinism	isoclinic class of #12463
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1675
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

248 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=egi,d^2=ghi,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fhi,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48415);