#146618

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

954 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=i,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	#72281262, #72317362, #72317371, …
maximal normal subgroup of	#72281262, #72317362, #72317371, …
maximal quotient of	C₂×#146618

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53814);