#500006

order	512
nilpotency class	4
isoclinism	isoclinic class of #500005
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23960
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

889 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)(4,10)(5,12)(6,30)(7,32)(8,14)(9,19)(11,28)(13,27)(16,37)(17,48)(18,46)(20,53)(21,54)(22,38)(23,57)(24,41)(25,44)(26,39)(29,42)(31,36)(33,34)(35,50)(40,56)(43,55)(45,49)(47,64)(51,60)(52,61)(58,63)(59,62);(3,16,18,25)(5,6)(7,27)(9,34,36,53)(11,42)(12,30)(13,32)(15,44,46,37)(17,47,49,59)(19,20,31,33)(21,55)(22,26)(23,56)(28,29)(35,60,61,63)(38,39)(40,57)(43,54)(45,64,48,62)(50,58,52,51);(1,3)(2,9)(4,17)(5,25)(6,31)(7,33)(8,18)(10,35)(11,34)(12,15)(13,44)(14,36)(16,29)(19,28)(20,27)(21,47)(22,50)(23,58)(24,49)(26,59)(30,53)(32,37)(38,63)(39,45)(40,64)(41,61)(42,46)(43,60)(48,54)(51,56)(52,55)(57,62);(1,4,8,24)(2,10,14,41)(3,17,18,49)(5,26,29,21)(6,22,28,55)(7,23,27,56)(9,35,36,61)(11,43,30,38)(12,39,42,54)(13,40,32,57)(15,45,46,48)(16,47,25,59)(19,52,31,50)(20,51,33,58)(34,60,53,63)(37,62,44,64)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250061, #72250066, #72250207, …
maximal normal subgroup of	#72250061, #72250066, #72250207, …
maximal quotient of	#72250626, #72281936

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407202);