#442255

order	512
nilpotency class	3
isoclinism	isoclinic class of #421447
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2474
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15706
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

720 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)(3,15,18,9,19,33,28,34)(4,20,23,37,24,36,29,10)(5,11,30,39,27,41,26,40)(16,31,50,42,21,48,44,59)(17,45,25,54,46,38,56,32)(22,47,52,62,53,61,57,35)(43,63,51,64,49,58,55,60);(1,3,8,18,7,19,6,28)(2,9,12,33,14,34,13,15)(4,21,23,44,24,16,29,50)(5,25,30,46,27,56,26,17)(10,31,20,42,37,48,36,59)(11,38,39,32,41,45,40,54)(22,51,52,49,53,55,57,43)(35,60,47,63,62,64,61,58);(1,4,6,23,7,24,8,29)(2,10,12,36,14,37,13,20)(3,16,18,44,19,21,28,50)(5,22,26,52,27,53,30,57)(9,31,15,42,34,48,33,59)(11,35,39,61,41,62,40,47)(17,43,25,55,46,49,56,51)(32,58,38,60,54,63,45,64);(1,5,6,26,7,27,8,30)(2,11,13,40,14,41,12,39)(3,17,18,25,19,46,28,56)(4,22,23,52,24,53,29,57)(9,32,33,45,34,54,15,38)(10,35,20,47,37,62,36,61)(16,43,44,55,21,49,50,51)(31,58,59,64,48,63,42,60)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 349451);