#6442595

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442595
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)(10,22)(16,32)(17,42)(18,38)(21,51)(23,39)(24,40)(25,37)(26,31)(27,34)(28,36)(29,44)(30,49)(33,53)(35,48)(41,46)(43,57)(45,50)(47,56)(52,59)(54,58)(55,61)(60,62)(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)(23,44)(24,49)(25,47)(27,45)(28,48)(29,34)(30,37)(33,55)(36,56)(39,41)(40,42)(51,60)(52,63)(53,62)(54,61)(57,59)(58,64);(1,4,7,22,8,23,6,27)(2,10,13,20,14,34,12,39)(3,16,19,46,18,44,26,45)(5,21,24,53,25,54,28,52)(9,29,15,41,31,32,38,50)(11,33,36,51,37,59,40,58)(17,43,49,62,47,61,48,63)(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)(22,53)(23,54)(26,48)(27,52)(29,55)(31,35)(32,57)(34,59)(38,56)(39,58)(41,60)(44,61)(45,63)(46,62)(50,64);(3,18)(9,31)(15,38)(16,44)(17,47)(19,26)(29,32)(30,35)(41,50)(42,56)(43,61)(45,46)(48,49)(55,57)(60,64)(62,63)

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,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249757, #72249764, #72249829, …
maximal normal subgroup of	#72249757, #72249764, #72249829, …
maximal quotient of	#72249958, #72322978

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349791);