#147986

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#394
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11042
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

665 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,e^b=ei,f^b=fi,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, 55182);