#494057

order	512
nilpotency class	4
isoclinism	isoclinic class of #33691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#20738
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

930 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^b=fi,d^c=dh,e^c=ei,f^c=fh,h^c=hi,f^d=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, 401253);