#132982

order	512
nilpotency class	4
isoclinism	isoclinic class of #132509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

504 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,f^c=fg,e^d=ei,f^d=fi,f^e=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, 40178);