#139180

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	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 287 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) = 128

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)(35,45)(36,49,51,41)(37,53)(38,44)(39,56)(40,43)(42,46)(47,60,59,61)(48,55)(50,64,54,63)(52,58,57,62);(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)(33,35,46,44,39,48,55,56,38,42,45,34,40,53,37,43)(36,47,61,49,52,63,50,62,51,59,60,41,57,64,54,58);(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)(33,36,39,52,38,51,40,57)(34,41,43,58,44,49,56,62)(35,47,48,63,42,59,53,64)(37,54,46,61,55,50,45,60)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=ghi,e^2=h,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, 46376);