#451916

order	512
nilpotency class	3
isoclinism	isoclinic class of #427666
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2527
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16021
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

743 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)(3,14,18,38)(4,20,23,9)(5,24)(6,11)(7,12)(8,17,31,19)(10,26)(13,28)(15,40)(16,45,46,30)(21,50,53,55)(22,34,54,33)(25,56,47,35)(27,36)(29,42)(32,52,49,64)(37,57)(39,44)(41,58)(43,48)(51,59)(60,61)(62,63);(1,3)(2,8)(4,15)(5,25)(6,19)(7,18)(9,29)(10,35)(11,38)(12,31)(13,14)(16,26)(17,28)(20,48)(21,51)(22,43)(23,44)(24,45)(27,47)(30,36)(32,59)(33,40)(34,39)(37,56)(41,52)(42,54)(46,57)(49,60)(50,58)(53,61)(55,63)(62,64);(1,4,6,22,7,23,28,54)(2,9,11,33,12,20,13,34)(3,15,17,42,18,44,19,43)(5,21,26,52,27,53,57,64)(8,29,14,39,31,48,38,40)(10,32,36,50,37,49,24,55)(16,41,25,51,46,62,47,61)(30,58,35,59,45,63,56,60);(1,5,6,26,7,27,28,57)(2,10,13,24,12,37,11,36)(3,16,17,25,18,46,19,47)(4,21,22,52,23,53,54,64)(8,30,38,56,31,45,14,35)(9,32,34,55,20,49,33,50)(15,41,42,51,44,62,43,61)(29,58,40,60,48,63,39,59)

Polycyclic presentation:

2^9|a^2=efgh,c^2=e,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,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, 359112);