#500354

order	512
nilpotency class	4
isoclinism	isoclinic class of #500353
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24089
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61290	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

707 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)(6,29)(7,30)(8,14)(9,19)(11,27)(13,26)(16,35)(17,48)(18,43)(20,41)(21,54)(22,36)(23,58)(24,39)(25,37)(28,40)(31,32)(33,50)(34,52)(38,57)(42,55)(44,47)(45,49)(46,63)(51,60)(53,61)(56,62)(59,64);(3,16,18,47)(5,6)(7,26)(9,32,34,41)(11,40)(12,29)(13,30)(15,44,43,35)(17,46,49,56)(19,20,52,31)(21,55)(22,25)(23,57)(27,28)(33,60,61,64)(36,37)(38,58)(42,54)(45,63,48,62)(50,59,53,51);(1,3,7,31,8,18,26,20)(2,9,13,44,14,34,30,35)(4,17,23,59,24,49,57,51)(5,16,27,52,28,47,6,19)(10,33,38,63,39,61,58,62)(11,41,12,43,29,32,40,15)(21,56,22,53,25,46,55,50)(36,60,54,45,42,64,37,48);(1,4,8,24)(2,10,14,39)(3,17,18,49)(5,25,28,21)(6,22,27,55)(7,23,26,57)(9,33,34,61)(11,42,29,36)(12,37,40,54)(13,38,30,58)(15,45,43,48)(16,46,47,56)(19,53,52,50)(20,51,31,59)(32,60,41,64)(35,62,44,63)

Polycyclic presentation:

2^9|b^2=e,c^2=eh,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=fhi,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250047, #72250066, #72250202, …
maximal normal subgroup of	#72250047, #72250066, #72250202, …
maximal quotient of	#72250461, #72279746, #72281818

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407550);