#6463583

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442595
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1675 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)(3,13,16,8)(4,9)(5,10)(6,12)(7,11)(14,27,42,39)(15,40,43,28)(17,38,25,29)(18,30,45,55)(19,31)(20,32)(21,33)(22,50,51,34)(23,36)(24,35)(26,37)(41,53,46,56)(44,54,61,63)(47,57)(48,58)(49,59)(52,60)(62,64);(1,3)(2,8)(4,18)(5,22)(6,25)(7,16)(9,30)(10,34)(11,13)(12,38)(14,21)(15,23)(17,26)(19,46)(20,45)(24,51)(27,33)(28,36)(29,37)(31,56)(32,55)(35,50)(39,59)(40,60)(41,47)(42,49)(43,52)(44,48)(53,57)(54,58)(61,62)(63,64);(1,4)(2,9)(3,14,16,42)(5,19)(6,21)(7,20)(8,27,13,39)(10,31)(11,32)(12,33)(15,41,43,46)(17,18,25,45)(22,44,51,61)(23,48)(24,47)(26,49)(28,53,40,56)(29,30,38,55)(34,54,50,63)(35,57)(36,58)(37,59)(52,62)(60,64);(1,5,6,23,7,24,26,52)(2,10,12,36,11,35,37,60)(3,15,17,22,16,43,25,51)(4,19,21,48,20,47,49,62)(8,28,29,34,13,40,38,50)(9,31,33,58,32,57,59,64)(14,41,18,44,42,46,45,61)(27,53,30,54,39,56,55,63);(2,11)(8,13)(9,32)(10,35)(12,37)(27,39)(28,40)(29,38)(30,55)(31,57)(33,59)(34,50)(36,60)(53,56)(54,63)(58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249769, #72249773, #72249829, …
maximal normal subgroup of	#72249769, #72249773, #72249829, …
maximal quotient of	#72250020, #72322900, #72322946

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370779);