#436656

order	512
nilpotency class	3
isoclinism	isoclinic class of #419458
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2709
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15567
prime radical	#23434	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1127 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)(3,15,19,41)(4,20,24,10)(5,11,27,39)(6,12)(7,14)(8,13)(9,18,33,29)(16,34)(17,37)(21,51)(22,35)(23,42,30,36)(25,57)(26,38,28,40)(31,46)(32,49)(43,47)(44,48)(45,59,53,63)(50,55)(52,60,64,58)(54,61)(56,62);(1,3)(2,9,13,33)(4,21)(5,25,27,49)(6,18)(7,29)(8,19)(10,34,20,43)(11,37)(12,15,14,41)(16,23)(17,26,48,28)(22,53,55,45)(24,46)(30,47)(31,42,51,36)(32,40)(35,60)(38,57)(39,44)(50,58)(52,54,64,56)(59,62)(61,63);(1,4)(2,10)(3,16)(5,22)(6,23)(7,30)(8,24)(9,31)(11,35)(12,36)(13,20)(14,42)(15,43)(17,45)(18,46)(19,47)(21,29)(25,52)(26,54)(27,55)(28,56)(32,58)(33,51)(34,41)(37,59)(38,61)(39,50)(40,62)(44,63)(48,53)(49,64)(57,60);(1,5,6,28,8,27,7,26)(2,11,14,38,13,39,12,40)(3,17,18,49,19,48,29,25)(4,22,23,56,24,55,30,54)(9,32,41,37,33,57,15,44)(10,35,42,61,20,50,36,62)(16,45,46,64,47,53,21,52)(31,58,34,59,51,60,43,63)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,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, 343852);