#449880

order	512
nilpotency class	3
isoclinism	isoclinic class of #420424
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2699
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15952
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

995 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357076);