#141364

order	512
nilpotency class	4
isoclinism	isoclinic class of #141346
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

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	#3169	C₂²
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

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

normal subgroups only

This group has 304 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) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=ghi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,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, 48560);