#139166

order	512
nilpotency class	4
isoclinism	isoclinic class of #139152
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#465
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

285 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,7,12,22)(4,14)(5,17)(6,9)(8,15)(10,19)(11,25,26,32)(13,29,28,21)(16,24)(18,23)(20,30,31,27)(33,34,38,44,40,45,39,43)(35,46,49,62,50,58,48,63)(36,51,52,42,37,54,53,57)(41,56,59,47,61,64,60,55);(1,3,6,12)(2,7)(4,11,16,26)(5,13,19,28)(8,20)(9,22)(10,21)(14,25)(15,27,18,30)(17,29)(23,31)(24,32)(33,35,37,55,39,49,52,56,40,50,36,47,38,48,53,64)(34,41,57,46,44,60,51,63,45,61,42,58,43,59,54,62);(1,4,6,16)(2,8,9,23)(3,11,12,26)(5,18,19,15)(7,20,22,31)(10,24,17,14)(13,30,28,27)(21,32,29,25)(33,36,39,53,40,37,38,52)(34,42,44,54,45,57,43,51)(35,47,49,64,50,55,48,56)(41,58,60,62,61,46,59,63)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46362);