#146615

order	512
nilpotency class	4
isoclinism	isoclinic class of #12753
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1790
commutator	C₂×D₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10745
prime radical	#3449	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,e^b=eh,f^b=fh,d^c=dhi,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281842, #72317364, #72317368, …
maximal normal subgroup of	#72281842, #72317364, #72317368, …
maximal quotient of	C₂×#146615

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53811);