#461832

order	512
nilpotency class	3
isoclinism	isoclinic class of #459145
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2751
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16404
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

733 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:

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

Polycyclic presentation:

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

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, 369028);