#419449

order	512
nilpotency class	3
isoclinism	isoclinic class of #419448
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:

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

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,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,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, 326645);