#132984

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

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

normal subgroups only

This group has 484 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) = 128

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

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

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,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, 40180);