#480437

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	#17363
prime radical	#3455	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,f^2=h,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,c^b=ci,f^b=fg,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387633);