#478743

order	512
nilpotency class	3
isoclinism	isoclinic class of #477321
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2830
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17165
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

642 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)(4,21)(5,27,31,42)(6,13)(7,12)(8,14)(9,18)(10,24)(11,30,41,29)(16,32)(17,47,51,33)(19,35)(20,34)(22,54)(23,37,57,59)(25,39)(26,38)(28,53,56,40)(36,45)(43,49,58,50)(44,62,63,61)(46,52)(48,64,55,60);(1,3,6,18,8,20,7,19)(2,9,12,15,14,35,13,34)(4,22,24,46,26,45,25,16)(5,17,29,49,31,51,30,50)(10,36,21,52,39,54,38,32)(11,33,27,58,41,47,42,43)(23,55,28,44,57,48,56,63)(37,62,40,60,59,61,53,64);(1,4,6,24,8,26,7,25)(2,10,12,21,14,39,13,38)(3,16,19,45,20,46,18,22)(5,28,29,57,31,56,30,23)(9,32,34,54,35,52,15,36)(11,40,27,59,41,53,42,37)(17,48,50,44,51,55,49,63)(33,61,43,60,47,62,58,64);(1,5)(2,11)(3,17)(4,23)(6,30)(7,29)(8,31)(9,33)(10,37)(12,42)(13,27)(14,41)(15,43)(16,44)(18,50)(19,49)(20,51)(21,53)(22,55)(24,56)(25,28)(26,57)(32,60)(34,58)(35,47)(36,62)(38,40)(39,59)(45,48)(46,63)(52,64)(54,61)

Polycyclic presentation:

2^9|a^2=fg,b^2=e,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^d=ei,f^d=fi,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, 385939);