#139172

order	512
nilpotency class	4
isoclinism	isoclinic class of #139144
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

294 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 84 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)(33,34,38,45,40,46,39,44)(35,47,50,57,52,63,51,62)(36,43,37,42,54,56,53,61)(41,55,59,64,60,48,58,49);(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)(33,35,37,49,38,50,54,55,40,52,53,64,39,51,36,48)(34,41,56,63,44,58,42,57,46,60,43,47,45,59,61,62);(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)(33,36,39,53,40,54,38,37)(34,42,45,56,46,61,44,43)(35,48,51,64,52,55,50,49)(41,57,59,63,60,62,58,47)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,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,d^b=dh,f^b=fh,e^c=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46368);