#474064

order	512
nilpotency class	3
isoclinism	isoclinic class of #460311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2500
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16895
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

879 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:

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

Polycyclic presentation:

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

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, 381260);