#141100

order	512
nilpotency class	4
isoclinism	isoclinic class of #125524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 329 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)(5,6,10,16,11,14,32,34,9,15,29,44,30,17,47,46)(7,18,22,49,23,43,54,45,21,12,33,36,20,37,31,60)(8,24,27,56,28,40,50,65,26,13,38,41,58,42,63,61)(19,35,53,62,51,39,64,66,52,48,59,68,25,55,57,67);(5,7,10,31,11,20,32,33,9,21,29,54,30,23,47,22)(6,12,34,49,17,37,16,45,15,18,46,36,14,43,44,60)(8,25,38,53,28,52,63,57,26,51,27,59,58,19,50,64)(13,39,65,66,40,48,56,68,24,55,61,67,42,35,41,62);(1,2,3,4)(5,8,11,28,9,26,30,58)(6,13,14,42,15,24,17,40)(7,19,23,51,21,52,20,25)(10,27,32,50,29,38,47,63)(12,35,37,39,18,48,43,55)(16,41,34,61,44,56,46,65)(22,53,54,64,33,59,31,57)(36,62,60,66,49,68,45,67)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=ehi,d^2=ghi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,d^b=dh,f^b=fhi,h^b=hi,d^c=di

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48296);