#501995

order	512
nilpotency class	4
isoclinism	isoclinic class of #501995
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

476 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)(4,23)(5,28)(6,13)(7,12)(8,14)(9,19)(10,26)(11,18)(16,40)(17,47)(20,43)(21,42)(22,44)(24,41)(25,58)(27,38)(29,31)(30,39)(32,57)(33,51)(34,45)(35,52)(36,61)(37,56)(46,59)(48,55)(49,54)(50,62)(53,63)(60,64);(1,3,16,45,7,21,5,19,8,22,18,51,6,20,29,52)(2,9,31,42,13,34,11,15,14,35,28,43,12,33,40,44)(4,24,48,61,26,50,56,64,27,53,46,49,30,17,25,57)(10,36,55,41,23,32,58,47,39,54,59,63,38,60,37,62);(1,4)(2,10)(3,17)(5,25)(6,30)(7,26)(8,27)(9,32)(11,37)(12,38)(13,23)(14,39)(15,41)(16,46)(18,48)(19,49)(20,53)(21,24)(22,50)(28,55)(29,56)(31,59)(33,36)(34,54)(35,60)(40,58)(42,62)(43,47)(44,63)(45,64)(51,57)(52,61);(1,3,18,51,7,21,29,52,8,22,16,45,6,20,5,19)(2,9,28,43,13,34,40,44,14,35,31,42,12,33,11,15)(4,17,48,57,26,24,56,61,27,50,46,64,30,53,25,49)(10,32,55,47,23,54,58,63,39,60,59,62,38,36,37,41)

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,d^b=di,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409191);