#139182

order	512
nilpotency class	4
isoclinism	isoclinic class of #139156
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 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

270 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 78 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)(33,34,39,36,38,43,40,51)(35,44,47,62,49,61,48,63)(37,50,54,57,53,52,55,42)(41,56,45,59,58,64,60,46);(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)(33,35,37,46,39,47,54,56,38,49,53,59,40,48,55,64)(34,41,57,61,51,60,50,62,43,58,42,44,36,45,52,63);(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)(33,36,40,43,38,51,39,34)(35,45,48,58,49,60,47,41)(37,52,55,42,53,50,54,57)(44,59,62,56,61,46,63,64)

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=ehi,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ehi,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46378);