#141268

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

278 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 90 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)(5,6,10,16,36,17,9,32,12,18,27,48,11,15,33,37)(7,19,23,50,21,41,34,39,25,13,22,40,24,46,52,47)(8,26,30,44,61,58,60,14,28,43,35,42,31,59,29,45)(20,51,54,38,55,65,62,68,56,64,57,49,53,67,63,66);(1,2,3,4)(5,7,10,34,36,24,9,23,12,25,27,52,11,21,33,22)(6,13,37,50,15,46,48,39,18,19,32,40,17,41,16,47)(8,20,35,63,61,53,29,57,28,56,30,62,31,55,60,54)(14,38,59,65,44,68,43,64,45,49,58,67,42,66,26,51);(5,8,27,30,36,61,33,60,12,28,10,35,11,31,9,29)(6,14,32,58,15,44,16,26,18,45,37,59,17,42,48,43)(7,20,52,62,24,53,22,54,25,56,34,63,21,55,23,57)(13,38,40,67,46,68,47,51,19,49,50,65,41,66,39,64)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=ehi,d^2=ghi,e^2=g,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,e^b=ei,f^b=fhi,h^b=hi,d^c=di,f^c=fi

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, 48464);