#133019

order	512
nilpotency class	4
isoclinism	isoclinic class of #132573
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

497 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,f^c=fgi,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, 40215);