#434687

order	512
nilpotency class	3
isoclinism	isoclinic class of #420416
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15506
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1029 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341883);