#460608

order	512
nilpotency class	3
isoclinism	isoclinic class of #460607
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16368
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

923 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)(3,15,19,42)(4,10)(5,26)(6,12)(7,13)(8,14)(9,18,33,20)(11,30)(16,34,46,43)(17,32,51,61)(21,52,47,31)(22,39)(23,36)(24,37)(25,38)(27,56)(28,40)(29,41)(35,55)(44,49,60,50)(45,58,57,64)(48,59,53,62)(54,63);(1,3,6,20,8,19,7,18)(2,9,12,42,14,33,13,15)(4,21,23,16,25,47,24,46)(5,17,30,49,29,51,28,50)(10,34,36,31,38,43,37,52)(11,32,41,60,40,61,26,44)(22,53,55,45,54,48,27,57)(35,62,63,58,56,59,39,64);(1,4,8,25)(2,10,14,38)(3,16,19,46)(5,27,29,55)(6,24,7,23)(9,31,33,52)(11,39,40,63)(12,37,13,36)(15,43,42,34)(17,48,51,53)(18,47,20,21)(22,30,54,28)(26,35,41,56)(32,59,61,62)(44,64,60,58)(45,50,57,49);(1,5)(2,11)(3,17)(4,22)(6,30)(7,28)(8,29)(9,32)(10,35)(12,41)(13,26)(14,40)(15,44)(16,45)(18,50)(19,51)(20,49)(21,53)(23,55)(24,27)(25,54)(31,58)(33,61)(34,62)(36,63)(37,39)(38,56)(42,60)(43,59)(46,57)(47,48)(52,64)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367804);