#146709

order	512
nilpotency class	4
isoclinism	isoclinic class of #12715
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1781
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10761
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ 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)(3,15,19,41)(4,21)(5,26,30,11)(6,31,33,12)(7,13)(8,14)(9,17)(10,23,39,57)(16,47,50,64)(18,36)(20,35,53,44)(22,56,28,40)(24,38)(25,55)(27,37)(29,43)(32,46)(34,60,63,51)(42,59)(45,58)(48,54)(49,61)(52,62);(1,3,7,20,8,19,29,53)(2,9,13,36,14,37,43,46)(4,16,24,51,25,50,59,60)(5,27,6,32,30,17,33,18)(10,34,40,64,39,63,56,47)(11,41,12,44,26,15,31,35)(21,54,38,45,55,62,42,61)(22,52,57,49,28,48,23,58);(1,4)(2,10)(3,16)(5,28)(6,23)(7,24)(8,25)(9,34)(11,42)(12,21)(13,40)(14,39)(15,45)(17,52)(18,49)(19,50)(20,51)(22,30)(26,38)(27,48)(29,59)(31,55)(32,58)(33,57)(35,62)(36,64)(37,63)(41,61)(43,56)(44,54)(46,47)(53,60)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317364, #72317395, #72317412
maximal normal subgroup of	#72317364, #72317395, #72317412

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53905);