#499179

order	512
nilpotency class	4
isoclinism	isoclinic class of #13442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2054
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2054
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1003 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)(3,16)(4,22)(5,12)(6,14)(7,13)(8,33)(9,15)(10,19)(11,26)(17,50)(18,37)(20,38)(21,49)(23,58)(24,43)(25,41)(27,61)(28,42)(29,57)(30,45)(31,36)(32,47)(34,46)(35,54)(39,55)(40,62)(44,60)(48,53)(51,56)(52,64)(59,63);(1,3)(2,10)(4,23)(5,18)(6,31)(7,19)(8,34)(9,21)(11,40)(12,36)(13,46)(14,16)(15,39)(17,27)(20,32)(22,56)(24,53)(25,54)(26,51)(28,62)(29,52)(30,55)(33,38)(35,43)(37,45)(41,58)(42,50)(44,48)(47,49)(57,63)(59,60)(61,64);(1,4,6,25,5,24,30,60,9,29,32,27,8,28,7,26)(2,11,13,42,33,61,47,57,15,44,45,43,12,41,14,22)(3,17,19,52,34,59,20,53,21,54,55,23,18,51,31,62)(10,35,16,48,36,63,37,64,39,50,49,40,38,56,46,58);(1,5,9,8)(2,12,15,33)(3,18,21,34)(4,24,29,28)(6,30,32,7)(10,36,39,38)(11,41,44,61)(13,14,45,47)(16,37,49,46)(17,51,54,59)(19,31,55,20)(22,43,57,42)(23,53,52,62)(25,60,27,26)(35,63,50,56)(40,58,48,64)

Polycyclic presentation:

2^9|c^2=e,d^2=i,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,h^a=hi,c^b=cg,e^b=eh,f^b=fh,g^b=gh,h^b=hi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320565, #72320615, #72320724
maximal normal subgroup of	#72320565, #72320615, #72320724
maximal quotient of	#72323017

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406375);