#146532

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,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, 53728);