#474443

order	512
nilpotency class	3
isoclinism	isoclinic class of #455830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16977
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

532 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381639);