#434719

order	512
nilpotency class	3
isoclinism	isoclinic class of #434718
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15506
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

1036 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)(3,13,16,8)(4,18,21,9)(5,23)(6,26)(7,11)(10,24)(12,29)(14,39)(15,42,43,55)(17,32,27,37)(19,30)(20,50,51,56)(22,34,53,47)(25,38)(28,36)(31,44,35,45)(33,52,46,54)(40,61)(41,58)(48,64)(49,60)(57,63)(59,62);(1,3)(2,8)(4,19)(5,15)(6,27)(7,16)(9,30)(10,35)(11,13)(12,37)(14,21)(17,29)(18,39)(20,48)(22,41)(23,55)(24,45)(25,43)(26,32)(28,44)(31,36)(33,59)(34,60)(38,42)(40,51)(46,57)(47,58)(49,53)(50,61)(52,62)(54,63)(56,64);(1,4)(2,9)(3,14,16,19)(5,20)(6,22)(7,21)(8,30,13,39)(10,33)(11,18)(12,34)(15,40,43,48)(17,41,27,49)(23,56)(24,52)(25,51)(26,47)(28,54)(29,53)(31,57,35,59)(32,58,37,60)(36,46)(38,50)(42,61,55,64)(44,62,45,63);(1,5,7,25)(2,10,11,36)(3,15,16,43)(4,20,21,51)(6,28,29,24)(8,31,13,35)(9,33,18,46)(12,38,26,23)(14,40,19,48)(17,45,27,44)(22,54,53,52)(30,57,39,59)(32,42,37,55)(34,50,47,56)(41,63,49,62)(58,61,60,64)

Polycyclic presentation:

2^9|a^2=fg,c^2=ef,d^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,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, 341915);