#141399

order	512
nilpotency class	4
isoclinism	isoclinic class of #141346
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1957
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#486
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

293 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eh,d^2=ghi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,d^b=dh,f^b=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48595);