#132073

order	512
nilpotency class	4
isoclinism	isoclinic class of #132072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8456
commutator	#117	C₂²×C₄
Frattini, cosocle	#405	C₂³
prime radical	#3539	C₄
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 544 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39269);