#478317

order	512
nilpotency class	3
isoclinism	isoclinic class of #476443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17140
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

778 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)(5,24,28,41)(6,30)(7,12)(8,13)(9,17)(10,23)(11,29,40,27)(15,45)(16,33,50,58)(18,34)(19,44)(21,39,56,63)(22,36)(25,43,51,38)(26,35,57,54)(31,37)(32,52)(42,48)(46,60,59,64)(47,53)(49,62,55,61);(1,3,6,17,8,19,7,18)(2,9,12,14,13,34,30,44)(4,15,22,47,23,48,31,52)(5,25,27,50,28,51,29,16)(10,32,36,42,20,53,37,45)(11,38,24,58,40,43,41,33)(21,55,26,46,56,49,57,59)(35,62,39,60,54,61,63,64);(1,4,6,22,8,23,7,31)(2,10,12,36,13,20,30,37)(3,15,17,47,19,48,18,52)(5,26,27,56,28,57,29,21)(9,32,14,42,34,53,44,45)(11,39,24,54,40,63,41,35)(16,49,25,59,50,55,51,46)(33,61,38,64,58,62,43,60);(1,5)(2,11)(3,16)(4,21)(6,29)(7,27)(8,28)(9,33)(10,35)(12,41)(13,40)(14,43)(15,46)(17,51)(18,25)(19,50)(20,54)(22,57)(23,56)(24,30)(26,31)(32,60)(34,58)(36,63)(37,39)(38,44)(42,62)(45,61)(47,55)(48,59)(49,52)(53,64)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385513);