#496950

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22135
prime radical	#3054	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

520 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=ceh,f^b=fi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404146);