#147391

order	512
nilpotency class	3
isoclinism	isoclinic class of #11115
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#388
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10922
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

677 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^c=ei,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, 54587);