#6573737

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573735
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4428 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)(4,9)(5,11)(6,12)(7,10)(13,43)(14,30)(15,29)(16,31)(17,49)(18,42)(19,34)(20,35)(21,33)(22,38)(23,40)(24,36)(25,41)(26,37)(27,39)(28,46)(32,44)(45,61)(47,55)(48,56)(50,64)(51,60)(52,59)(53,58)(54,63)(57,62);(1,3)(2,8)(4,17)(5,22)(6,25)(7,15)(9,32)(10,29)(11,38)(12,41)(13,21)(14,23)(16,26)(18,46)(19,45)(20,44)(24,48)(27,47)(28,33)(30,40)(31,37)(34,54)(35,49)(36,56)(39,55)(42,43)(50,51)(52,63)(53,62)(57,60)(58,64)(59,61);(1,4)(2,9)(3,13)(5,19)(6,18)(7,20)(8,28)(10,35)(11,34)(12,42)(14,45)(15,46)(16,44)(17,25)(21,26)(22,50)(23,53)(24,52)(27,51)(29,43)(30,54)(31,49)(32,41)(33,37)(36,59)(38,57)(39,60)(40,58)(47,63)(48,62)(55,61)(56,64);(2,10)(4,18)(5,23)(6,26)(8,29)(9,33)(11,39)(13,44)(14,22)(16,25)(17,46)(19,52)(20,21)(24,27)(28,32)(30,56)(35,42)(36,40)(38,55)(43,49)(45,63)(47,48)(57,64)(58,60);(1,5,7,24)(2,11,10,36)(3,14,15,47)(4,19,20,52)(6,27,26,23)(8,30,29,55)(9,34,35,59)(12,39,37,40)(13,45,46,63)(16,48,25,22)(17,50,44,62)(18,51,21,53)(28,54,43,61)(31,56,41,38)(32,57,49,64)(33,58,42,60)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312282, #72312361, #72312376, …
maximal normal subgroup of	#72312282, #72312361, #72312376, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480933);