#504676

order	512
nilpotency class	4
isoclinism	isoclinic class of #504676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26801
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1765 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

497 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)(3,15)(4,21)(5,11)(6,12)(7,29)(8,14)(9,17)(10,24)(13,32)(16,46)(18,52)(19,36)(20,34)(22,37)(23,38)(25,44)(26,39)(27,42)(28,41)(30,35)(31,60)(33,50)(40,47)(43,54)(45,49)(48,61)(51,62)(53,55)(56,59)(57,63)(58,64);(1,3,5,17,8,20,6,19)(2,9,12,15,14,36,11,34)(4,22,23,53,26,57,24,58)(7,30,27,18,32,51,28,54)(10,37,39,64,38,63,21,55)(13,43,42,62,29,52,41,35)(16,47,49,59,50,31,48,25)(33,44,45,60,46,56,61,40);(1,4,5,23,8,26,6,24)(2,10,12,39,14,38,11,21)(3,16,17,49,20,50,19,48)(7,31,27,25,32,47,28,59)(9,33,15,45,36,46,34,61)(13,44,42,60,29,56,41,40)(18,53,51,57,54,58,30,22)(35,55,43,37,62,64,52,63);(2,11,14,12)(4,23,26,24)(7,32)(9,34,36,15)(10,38)(13,42,29,41)(16,48,50,49)(18,54)(21,39)(22,53,57,58)(25,31,59,47)(27,28)(30,51)(35,43,62,52)(37,63)(55,64)

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=fh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=gh,h^c=hi,g^d=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411872);