#501994

order	512
nilpotency class	4
isoclinism	isoclinic class of #501994
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3341
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#25032
prime radical	#33617	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

478 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)(4,21)(5,27)(6,30)(7,12)(8,13)(9,17)(10,24)(11,15)(16,45)(18,52)(19,41)(20,42)(22,55)(23,35)(25,37)(26,36)(28,38)(29,59)(31,60)(32,50)(33,51)(34,47)(39,48)(40,43)(44,54)(46,62)(49,63)(53,57)(56,58)(61,64);(1,3,15,43,6,18,29,51,8,20,28,50,7,19,5,17)(2,9,27,41,12,32,38,42,13,33,59,52,30,40,11,14)(4,22,56,47,25,49,23,57,26,48,44,64,24,16,46,60)(10,34,54,55,36,31,35,45,37,61,58,39,21,53,62,63);(1,4)(2,10)(3,16)(5,23)(6,25)(7,24)(8,26)(9,31)(11,35)(12,36)(13,37)(14,39)(15,44)(17,47)(18,22)(19,48)(20,49)(21,30)(27,58)(28,56)(29,46)(32,61)(33,53)(34,40)(38,62)(41,63)(42,55)(43,57)(45,52)(50,60)(51,64)(54,59);(1,3,15,43,6,18,29,51,8,20,28,50,7,19,5,17)(2,9,27,41,12,32,38,42,13,33,59,52,30,40,11,14)(4,16,44,57,25,22,46,64,26,49,56,60,24,48,23,47)(10,31,58,63,36,61,62,55,37,53,54,45,21,34,35,39)

Polycyclic presentation:

2^9|b^2=ef,d^2=eh,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409190);