#6442597

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442597
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2408
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4425 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)(3,15)(4,20)(5,11)(6,13)(7,12)(8,14)(9,19)(10,23)(16,32)(17,42)(18,38)(21,51)(22,39)(24,40)(25,37)(26,31)(27,34)(28,36)(29,46)(30,49)(33,54)(35,48)(41,45)(43,57)(44,50)(47,56)(52,59)(53,58)(55,62)(60,61)(63,64);(1,3)(2,9)(4,16)(5,17)(6,26)(7,19)(8,18)(10,32)(11,35)(12,38)(13,15)(14,31)(20,50)(21,43)(22,46)(23,45)(24,49)(25,47)(27,44)(28,48)(29,34)(30,37)(33,55)(36,56)(39,41)(40,42)(51,60)(52,62)(53,63)(54,61)(57,58)(59,64);(1,4,7,23,8,22,6,27)(2,10,13,20,14,34,12,39)(3,16,19,45,18,46,26,44)(5,21,24,54,25,52,28,53)(9,29,15,41,31,32,38,50)(11,33,36,51,37,58,40,59)(17,43,49,61,47,62,48,63)(30,55,42,60,35,57,56,64);(1,5)(2,11)(3,17)(4,21)(6,28)(7,24)(8,25)(9,30)(10,33)(12,40)(13,36)(14,37)(15,42)(16,43)(18,47)(19,49)(20,51)(22,52)(23,54)(26,48)(27,53)(29,55)(31,35)(32,57)(34,58)(38,56)(39,59)(41,60)(44,63)(45,61)(46,62)(50,64);(3,18)(4,22)(9,31)(10,34)(15,38)(17,47)(19,26)(20,39)(21,52)(23,27)(30,35)(33,58)(42,56)(48,49)(51,59)(53,54)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285642, #72285753, #72286119, …
maximal normal subgroup of	#72285642, #72285753, #72286119, …
maximal quotient of	C₂×#6442597

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349793);