#146536

order	512
nilpotency class	4
isoclinism	isoclinic class of #12873
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1823
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

949 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)(3,15,18,42)(4,20,22,10)(5,25)(6,14)(7,13)(8,12)(9,17,32,47)(11,28)(16,43)(19,48,30,37)(21,51,53,57)(23,35,24,36)(26,33,58,40)(27,38)(29,39)(31,44)(34,55,50,54)(41,46)(45,60)(49,63)(52,61)(56,62)(59,64);(1,3,6,19,8,18,7,30)(2,9,13,33,12,32,14,40)(4,16,23,45,22,46,24,56)(5,26,27,47,29,58,28,17)(10,31,35,59,20,49,36,61)(11,37,39,42,38,48,25,15)(21,52,54,63,53,64,55,44)(34,60,51,43,50,62,57,41);(1,4)(2,10)(3,16)(5,21)(6,23)(7,24)(8,22)(9,31)(11,34)(12,20)(13,35)(14,36)(15,41)(17,44)(18,46)(19,45)(25,57)(26,52)(27,54)(28,55)(29,53)(30,56)(32,49)(33,59)(37,60)(38,50)(39,51)(40,61)(42,43)(47,63)(48,62)(58,64)

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,f^b=fi,h^b=hi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285832, #72287540, #72307171, …
maximal normal subgroup of	#72285832, #72287540, #72307171, …
maximal quotient of	#72285955, #72287578

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53732);