#146544

order	512
nilpotency class	4
isoclinism	isoclinic class of #126723
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1736
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10730
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

951 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)(3,15,18,44)(4,20,22,10)(5,25)(6,14)(7,13)(8,12)(9,17,33,49)(11,29)(16,43)(19,50,31,38)(21,52,27,59)(23,36,24,37)(26,34,58,42)(28,40)(30,41)(32,45)(35,55,39,54)(46,60)(47,62)(48,51)(53,63)(56,64)(57,61);(1,3,6,19,8,18,7,31)(2,9,13,34,12,33,14,42)(4,16,23,47,22,46,24,56)(5,26,28,49,30,58,29,17)(10,32,36,61,20,51,37,63)(11,38,41,44,40,50,25,15)(21,53,54,48,27,57,55,45)(35,62,52,43,39,64,59,60);(1,4)(2,10)(3,16)(5,27)(6,23)(7,24)(8,22)(9,32)(11,39)(12,20)(13,36)(14,37)(15,43)(17,48)(18,46)(19,47)(21,30)(25,52)(26,57)(28,55)(29,54)(31,56)(33,51)(34,61)(35,40)(38,64)(41,59)(42,63)(44,60)(45,49)(50,62)(53,58)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287540, #72307180, #72307189, …
maximal normal subgroup of	#72287540, #72307180, #72307189, …
maximal quotient of	#72287583, #72318976

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53740);