#6433174

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433173
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2622
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60368	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2250 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,f^b=fi,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306351, #72306460, #72306465
maximal normal subgroup of	#72306351, #72306460, #72306465

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340370);