#132988

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	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 486 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:

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

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=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, 40184);