#499082

order	512
nilpotency class	4
isoclinism	isoclinic class of #498968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#486
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2041
prime radical	#60643	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1175 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)(3,14)(4,19)(5,26)(6,31)(7,34)(8,11)(9,16)(10,22)(12,30)(13,28)(15,51)(17,40)(18,46)(20,49)(21,58)(23,42)(24,48)(25,57)(27,38)(29,45)(32,47)(33,44)(35,50)(36,43)(37,55)(39,56)(41,53)(52,62)(54,64)(59,61)(60,63);(1,3)(2,9)(4,20)(5,27)(6,32)(7,35)(8,18)(10,41)(11,38)(12,46)(13,47)(14,26)(15,23)(16,30)(17,29)(19,37)(21,59)(22,54)(24,55)(25,60)(28,56)(31,40)(33,52)(34,39)(36,53)(42,62)(43,51)(44,64)(45,50)(48,63)(49,58)(57,61);(1,4,5,21,8,25,30,24)(2,10,12,43,11,42,26,44)(3,15,16,52,18,54,27,53)(6,33,28,23,7,36,29,22)(9,37,14,49,38,61,46,63)(13,48,31,57,45,58,34,19)(17,55,35,60,56,59,32,20)(39,51,50,41,40,64,47,62);(2,11)(9,38)(10,42)(12,26)(13,45)(14,46)(19,57)(31,34)(37,61)(39,40)(41,62)(43,44)(47,50)(48,58)(49,63)(51,64)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249769, #72249776, #72249835, …
maximal normal subgroup of	#72249769, #72249776, #72249835, …
maximal quotient of	#72250022, #72323097

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406278);