#149105

order	512
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#397
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11298
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

423 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56301);