#480321

order	512
nilpotency class	4
isoclinism	isoclinic class of #480315
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1464
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #117 ◁ #2873 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

705 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=i,f^2=h,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=ei,f^b=fgi,d^c=di,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, 387517);