#479797

order	512
nilpotency class	3
isoclinism	isoclinic class of #477835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2793
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17254
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

601 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)(3,14,20,42)(4,21)(5,10,28,41)(6,11)(7,13)(8,18,32,19)(9,24)(12,15)(16,33,22,30)(17,44)(23,55,56,38)(25,35)(26,34,57,62)(27,39,29,40)(31,50)(36,45)(37,51)(43,47,53,48)(46,58)(49,59)(52,60)(54,61)(63,64);(1,3,15,18,7,20,6,19)(2,8,12,14,13,32,11,42)(4,22,24,47,25,16,45,48)(5,17,29,50,28,52,27,51)(9,33,35,43,36,30,21,53)(10,37,40,44,41,31,39,60)(23,54,26,49,56,46,57,64)(34,58,38,63,62,61,55,59);(1,4,6,24,7,25,15,45)(2,9,12,21,13,36,11,35)(3,16,19,48,20,22,18,47)(5,26,27,56,28,57,29,23)(8,30,14,43,32,33,42,53)(10,38,40,34,41,55,39,62)(17,49,51,46,52,64,50,54)(31,59,60,61,37,63,44,58);(1,5)(2,10)(3,17)(4,23)(6,29)(7,28)(8,31)(9,34)(11,40)(12,39)(13,41)(14,44)(15,27)(16,46)(18,51)(19,50)(20,52)(21,38)(22,54)(24,57)(25,56)(26,45)(30,58)(32,37)(33,61)(35,55)(36,62)(42,60)(43,63)(47,64)(48,49)(53,59)

Polycyclic presentation:

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

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, 386993);