#452085

order	512
nilpotency class	3
isoclinism	isoclinic class of #418871
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2697
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16028
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

761 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)(3,13,17,8)(4,18)(5,22,24,10)(6,25)(7,11)(9,21)(12,27)(14,38,40,56)(15,34)(16,30,45,37)(19,47,49,46)(20,36)(23,35,54,52)(26,33)(28,41,31,44)(29,43)(32,50,51,53)(39,60)(42,57)(48,64)(55,62)(58,61)(59,63);(1,3,6,16,7,17,27,45)(2,8,25,37,11,13,12,30)(4,14,26,44,20,40,21,41)(5,15,23,42,24,43,54,63)(9,31,18,38,33,28,36,56)(10,34,52,57,22,29,35,59)(19,48,53,61,49,39,50,62)(32,58,46,60,51,55,47,64);(1,4)(2,9)(3,14)(5,19)(6,26)(7,20)(8,28)(10,32)(11,33)(12,36)(13,31)(15,39)(16,44)(17,40)(18,25)(21,27)(22,51)(23,53)(24,49)(29,55)(30,38)(34,58)(35,47)(37,56)(41,45)(42,62)(43,48)(46,52)(50,54)(57,60)(59,64)(61,63);(1,5,6,23,7,24,27,54)(2,10,25,52,11,22,12,35)(3,15,16,42,17,43,45,63)(4,19,26,53,20,49,21,50)(8,29,37,59,13,34,30,57)(9,32,18,46,33,51,36,47)(14,39,44,62,40,48,41,61)(28,55,56,64,31,58,38,60)

Polycyclic presentation:

2^9|a^2=gh,b^2=ef,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,h^b=hi,e^c=ei,f^c=fi,g^d=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, 359281);