#141218

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eg,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, 48414);