#478527

order	512
nilpotency class	3
isoclinism	isoclinic class of #478526
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17152
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

673 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)(3,15,19,9)(4,20,25,38)(5,26,29,42)(6,13)(7,14)(8,12)(10,23,37,24)(11,28,41,30)(16,35)(17,49)(18,43,31,34)(21,44)(22,36)(27,45)(32,47)(33,50)(39,51)(40,57)(46,58,54,60)(48,59)(52,56)(53,61,63,64)(55,62);(1,3,6,18,8,19,7,31)(2,9,13,34,12,15,14,43)(4,21,23,16,25,47,24,48)(5,27,28,51,29,50,30,17)(10,35,20,32,37,59,38,44)(11,39,26,33,41,49,42,45)(22,54,55,63,57,46,56,53)(36,58,62,61,40,60,52,64);(1,4)(2,10,12,37)(3,16)(5,22)(6,23)(7,24)(8,25)(9,32,15,44)(11,40,41,36)(13,20,14,38)(17,46)(18,47)(19,48)(21,31)(26,52,42,62)(27,53)(28,55)(29,57)(30,56)(33,60,45,58)(34,59,43,35)(39,61,49,64)(50,63)(51,54);(1,5,8,29)(2,11)(3,17,19,51)(4,22,25,57)(6,30,7,28)(9,33)(10,36)(12,41)(13,42)(14,26)(15,45)(16,46,48,54)(18,50,31,27)(20,52)(21,53,47,63)(23,56,24,55)(32,58)(34,39)(35,61)(37,40)(38,62)(43,49)(44,60)(59,64)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385723);