#464657

order	512
nilpotency class	3
isoclinism	isoclinic class of #464656
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16508
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1463 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264928, #72264954, #72265041
maximal normal subgroup of	#72264928, #72264954, #72265041
maximal quotient of	#72313945

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371853);