#6432429

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431743
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#127
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂²×#399
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3033 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285720, #72285784, #72286055, …
maximal normal subgroup of	#72285720, #72285784, #72286055, …
maximal quotient of	C₂×#6432429

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339625);