#6349802

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349761
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#62761	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2591 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306666, #72306667, #72306683, …
maximal normal subgroup of	#72306666, #72306667, #72306683, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256998);