#419457

order	512
nilpotency class	3
isoclinism	isoclinic class of #419456
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15270
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1288 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326653);