#6442739

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442596
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

4455 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)(3,8)(4,17)(5,10)(6,23)(7,12)(9,20)(11,24)(13,35)(14,26)(15,27)(16,42)(18,44)(19,45)(21,33)(22,50)(25,38)(28,41)(29,53)(30,47)(31,48)(32,39)(34,49)(36,57)(37,43)(40,56)(46,62)(51,58)(52,60)(54,64)(55,63)(59,61);(1,3)(2,8)(4,13)(5,14)(6,16)(7,15)(9,29)(10,32)(11,28)(12,27)(17,43)(18,36)(19,37)(20,38)(21,39)(22,40)(23,42)(24,41)(25,31)(26,33)(30,51)(34,56)(35,45)(44,57)(46,59)(47,58)(48,53)(49,60)(50,52)(54,55)(61,62)(63,64);(1,4,6,20,7,19,24,48)(2,9,11,17,12,31,23,45)(3,13,16,38,15,37,41,53)(5,18,22,47,21,46,49,63)(8,25,28,35,27,29,42,43)(10,30,34,44,33,55,50,62)(14,36,40,58,39,59,60,64)(26,51,52,57,32,54,56,61);(1,5)(2,10)(3,14)(4,18)(6,22)(7,21)(8,26)(9,30)(11,34)(12,33)(13,36)(15,39)(16,40)(17,44)(19,46)(20,47)(23,50)(24,49)(25,51)(27,32)(28,52)(29,54)(31,55)(35,57)(37,59)(38,58)(41,60)(42,56)(43,61)(45,62)(48,63)(53,64);(3,15)(4,19)(8,27)(9,31)(14,39)(16,41)(17,45)(18,46)(20,48)(26,32)(28,42)(30,55)(40,60)(44,62)(47,63)(52,56)

Polycyclic presentation:

2^9|c^2=g,g^2=i,b^a=bf,c^a=cg,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,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	#72303777, #72303799, #72303806, …
maximal normal subgroup of	#72303777, #72303799, #72303806, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349935);