#146560

order	512
nilpotency class	4
isoclinism	isoclinic class of #12750
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

945 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)(3,14,18,42)(4,20,23,35)(5,25)(6,11)(7,13)(8,16,32,48)(9,22,34,24)(10,27)(12,29)(15,43)(17,41,19,36)(21,52,55,33)(26,39,49,31)(28,37)(30,58)(38,57)(40,46)(44,59)(45,63)(47,62)(50,64)(51,54,61,56)(53,60);(1,3,6,17,7,18,29,19)(2,8,11,31,13,32,12,39)(4,15,24,46,23,47,22,45)(5,26,27,48,28,49,57,16)(9,30,35,59,34,60,20,50)(10,36,37,14,38,41,25,42)(21,53,56,64,55,58,54,44)(33,43,61,40,52,62,51,63);(1,4)(2,9)(3,15)(5,21)(6,24)(7,23)(8,30)(10,33)(11,35)(12,20)(13,34)(14,40)(16,44)(17,46)(18,47)(19,45)(22,29)(25,51)(26,53)(27,56)(28,55)(31,59)(32,60)(36,43)(37,61)(38,52)(39,50)(41,62)(42,63)(48,64)(49,58)(54,57)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,d^b=dhi,f^b=fi,h^b=hi,e^c=ei,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, 53756);