#494523

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1879
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1283 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fhi,g^c=ghi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314256, #72314275
maximal normal subgroup of	#72314256, #72314275

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401719);