#146706

order	512
nilpotency class	4
isoclinism	isoclinic class of #12725
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10761
prime radical	#3449	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

991 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)(4,18)(5,23,25,10)(6,27)(7,12)(8,15,30,24)(9,20,33,19)(11,28)(14,29)(16,46)(17,35)(21,52)(22,32)(26,36,44,37)(31,54,55,45)(34,50,56,51)(38,41,48,40)(39,47)(42,62)(43,58)(49,53)(57,63)(59,60,64,61);(1,3,6,16,7,17,28,47)(2,8,11,31,12,30,27,55)(4,14,21,42,22,43,53,63)(5,24,26,54,25,15,44,45)(9,29,34,57,33,58,56,62)(10,35,36,46,23,13,37,39)(18,48,49,64,32,38,52,59)(19,41,50,61,20,40,51,60);(1,4)(2,9)(3,14)(5,20)(6,21)(7,22)(8,29)(10,18)(11,34)(12,33)(13,38)(15,41)(16,42)(17,43)(19,25)(23,32)(24,40)(26,51)(27,56)(28,53)(30,58)(31,57)(35,48)(36,49)(37,52)(39,59)(44,50)(45,61)(46,64)(47,63)(54,60)(55,62)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53902);