#133052

order	512
nilpotency class	4
isoclinism	isoclinic class of #132549
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

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

normal subgroups only

This group has 506 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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,c^b=ci,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,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, 40248);