#375719

order	512
nilpotency class	3
isoclinism	isoclinic class of #375718
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14169
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1061 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)(3,14,18,30)(4,9)(5,24,26,37)(6,11)(7,13)(8,17,31,47)(10,25,36,55)(12,27)(15,39,43,32)(16,44)(19,48,42,28)(20,50,52,54)(21,34)(22,38)(23,35)(29,46)(33,51,59,53)(40,61)(41,56)(45,57)(49,60)(58,62)(63,64);(1,3,7,18)(2,8)(4,19)(5,16,26,46)(6,17,27,47)(9,32,35,39)(10,29)(11,14)(12,30)(13,31)(15,21)(20,49)(22,43)(23,42)(24,40)(25,45,55,61)(28,34,48,38)(33,58,59,60)(36,44)(37,57)(41,51)(50,64,54,56)(52,62)(53,63);(1,4,7,23)(2,9,13,35)(3,15)(5,20,26,52)(6,22,27,21)(8,28)(10,33,36,59)(11,38,12,34)(14,39)(16,41)(17,19)(18,43)(24,54,37,50)(25,53,55,51)(29,56)(30,32)(31,48)(40,60)(42,47)(44,64)(45,49)(46,63)(57,58)(61,62);(1,5,6,25,7,26,27,55)(2,10,12,37,13,36,11,24)(3,16,17,45,18,46,47,61)(4,20,22,53,23,52,21,51)(8,29,30,57,31,44,14,40)(9,33,34,50,35,59,38,54)(15,41,19,49,43,63,42,62)(28,56,32,58,48,64,39,60)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264954, #72264989
maximal normal subgroup of	#72264954, #72264989
maximal quotient of	#72314255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282915);