#147553

order	512
nilpotency class	3
isoclinism	isoclinic class of #2008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#392
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10964
prime radical	C₂×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

993 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=di,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, 54749);