#497763

order	512
nilpotency class	3
isoclinism	isoclinic class of #360368
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2344
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22603
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

485 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404959);