#146543

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

971 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)(3,13,16,41)(4,18,20,9)(5,22)(6,27)(7,11)(8,15,31,46)(10,25)(12,29)(14,40)(17,47,28,35)(19,50,24,57)(21,34,53,49)(23,32,56,38)(26,39)(30,42)(33,52,36,58)(37,59)(43,60)(44,62)(45,48)(51,63)(54,64)(55,61);(1,3)(2,8,11,31)(4,14)(5,23)(6,28)(7,16)(9,30,18,48)(10,35,37,47)(12,38,27,32)(13,39,41,22)(15,25)(17,29)(19,51)(20,43)(21,54)(24,55)(26,56)(33,62,36,64)(34,63,49,61)(40,57,60,50)(42,52)(44,53)(45,58)(46,59);(1,4)(2,9)(3,14)(5,24)(6,21)(7,20)(8,30)(10,36)(11,18)(12,34)(13,40)(15,45)(16,43)(17,44)(19,26)(22,50)(23,55)(25,58)(27,49)(28,54)(29,53)(31,48)(32,61)(33,37)(35,64)(38,63)(39,57)(41,60)(42,46)(47,62)(51,56)(52,59)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,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	#72280256, #72280507, #72281021, …
maximal normal subgroup of	#72280256, #72280507, #72281021, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53739);