#461792

order	512
nilpotency class	3
isoclinism	isoclinic class of #458069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2653
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16402
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

741 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)(4,20)(5,25,27,35)(6,28)(7,12)(8,13)(9,17)(10,22)(11,26,36,29)(15,40)(16,45,47,56)(18,32)(19,39)(21,33,54,60)(23,34)(24,52)(30,44)(31,49,38,48)(37,43)(41,59,46,58)(42,50)(51,53,57,55)(61,62,64,63);(1,3,6,18,8,17,7,19)(2,9,12,32,13,14,28,39)(4,15,22,43,24,42,23,44)(5,16,26,48,27,47,29,49)(10,30,20,50,34,37,52,40)(11,31,35,45,36,38,25,56)(21,41,53,62,54,46,55,63)(33,58,51,64,60,59,57,61);(1,4,6,22,8,24,7,23)(2,10,12,20,13,34,28,52)(3,15,18,43,17,42,19,44)(5,21,26,53,27,54,29,55)(9,30,32,50,14,37,39,40)(11,33,35,51,36,60,25,57)(16,46,48,63,47,41,49,62)(31,59,45,61,38,58,56,64);(1,5)(2,11)(3,16)(4,21)(6,26)(7,29)(8,27)(9,31)(10,33)(12,35)(13,36)(14,38)(15,41)(17,47)(18,48)(19,49)(20,51)(22,53)(23,55)(24,54)(25,28)(30,58)(32,45)(34,60)(37,59)(39,56)(40,61)(42,46)(43,62)(44,63)(50,64)(52,57)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368988);