#146534

order	512
nilpotency class	4
isoclinism	isoclinic class of #12741
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1786
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

941 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53730);