#498863

order	512
nilpotency class	4
isoclinism	isoclinic class of #498863
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#493
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2011
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1537 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)(3,15)(4,21)(5,12)(6,11)(7,31)(8,13)(9,18)(10,23)(14,27)(16,50)(17,49)(19,37)(20,36)(22,47)(24,44)(25,59)(26,46)(28,45)(29,41)(30,43)(32,42)(33,39)(34,57)(35,55)(38,54)(40,60)(48,53)(51,63)(52,58)(56,62)(61,64);(1,3)(2,9)(4,16)(5,26)(6,18)(7,32)(8,17)(10,38)(11,42)(12,15)(13,35)(14,46)(19,28)(20,27)(21,58)(22,53)(23,61)(24,57)(25,51)(29,56)(30,55)(31,37)(33,54)(34,40)(36,43)(39,48)(41,63)(44,62)(45,49)(47,50)(52,60)(59,64);(1,4)(2,10)(3,16)(5,23)(6,29)(7,33)(8,25)(9,34)(11,21)(12,44)(13,40)(14,47)(15,48)(17,51)(18,56)(19,52)(20,57)(22,30)(24,27)(26,61)(28,60)(31,41)(32,54)(35,38)(36,58)(37,50)(39,45)(42,64)(43,59)(46,63)(49,62)(53,55);(3,17)(9,35)(15,49)(16,51)(18,55)(19,26)(20,32)(34,38)(36,42)(37,46)(48,62)(50,63)(52,61)(53,56)(54,57)(58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249764, #72249769, #72249835, …
maximal normal subgroup of	#72249764, #72249769, #72249835, …
maximal quotient of	#72250021, #72323021

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406059);