#6437951

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437950
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2410
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4464 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312303, #72312423
maximal normal subgroup of	#72312303, #72312423

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345147);