#497182

order	512
nilpotency class	3
isoclinism	isoclinic class of #380391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2331
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22311
prime radical	C₂³	#568
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

352 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=ef,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=ceh,f^b=fi,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404378);