#132068

order	512
nilpotency class	4
isoclinism	isoclinic class of #132067
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	#62274	C₂
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 553 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:

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

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,h^a=hi,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,e^d=ei,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39264);