#6573702

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573699
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

4398 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)(3,14)(4,10)(5,11)(6,27)(7,12)(8,13)(9,15)(16,43)(17,44)(18,30)(19,33)(20,34)(21,35)(22,37)(23,32)(24,39)(25,41)(26,42)(28,36)(29,40)(31,45)(38,49)(46,54)(47,57)(48,61)(50,55)(51,58)(52,59)(53,60)(56,62)(63,64);(1,3)(2,9)(4,16)(5,23)(6,19)(7,15)(8,18)(10,31)(11,38)(12,30)(13,33)(14,27)(17,25)(20,50)(21,48)(22,46)(24,47)(26,51)(28,45)(29,49)(32,41)(34,43)(35,57)(36,54)(37,55)(39,56)(40,58)(42,44)(52,63)(53,62)(59,61)(60,64);(3,15)(4,20)(6,7)(9,30)(10,34)(12,27)(14,33)(16,46)(17,23)(18,19)(21,52)(22,28)(24,53)(25,29)(31,55)(32,38)(35,59)(36,37)(39,60)(40,41)(44,58)(47,63)(49,51)(56,61);(1,4,8,22)(2,10,13,37)(3,16,18,46)(5,24,26,52)(6,20,7,28)(9,31,33,55)(11,39,42,59)(12,36,27,34)(14,43,30,54)(15,45,19,50)(17,48,49,62)(21,29,53,25)(23,47,51,63)(32,57,58,64)(35,40,60,41)(38,56,44,61);(1,5,8,26)(2,11,13,42)(3,17,18,49)(4,21,22,53)(6,25,7,29)(9,32,33,58)(10,35,37,60)(12,40,27,41)(14,44,30,38)(15,23,19,51)(16,47,46,63)(20,52,28,24)(31,56,55,61)(34,59,36,39)(43,57,54,64)(45,62,50,48)

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,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	#72312286, #72312303, #72312372, …
maximal normal subgroup of	#72312286, #72312303, #72312372, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480898);