#354315

order	512
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#386
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	C₄×#386
prime radical	#3449	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1769 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281077, #72317339, #72317342, …
maximal normal subgroup of	#72281077, #72317339, #72317342, …
maximal quotient of	C₂×#354315

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261511);