#6437631

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437629
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2403
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

5991 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)(5,10)(6,11)(7,12)(13,28)(14,25)(15,26)(16,27)(17,29)(18,30)(19,31)(20,33)(21,34)(22,35)(23,36)(24,39)(32,41)(37,53)(38,51)(40,50)(42,49)(43,52)(44,54)(45,55)(46,57)(47,58)(48,60)(56,62)(59,64)(61,63);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,28)(10,32)(11,26)(12,27)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,31)(25,34)(29,48)(30,51)(33,56)(35,49)(36,52)(44,59)(45,60)(46,61)(47,62)(50,58)(53,55)(54,63)(57,64);(1,4,7,19)(2,9,12,31)(3,13,16,39)(5,17,21,45)(6,22,23,18)(8,24,27,28)(10,29,34,55)(11,35,36,30)(14,37,41,60)(15,42,43,38)(20,46,47,44)(25,48,32,53)(26,51,52,49)(33,57,58,54)(40,61,62,59)(50,64,56,63);(1,5)(2,10)(3,14)(4,17)(6,20)(7,21)(8,25)(9,29)(11,33)(12,34)(13,37)(15,40)(16,41)(18,44)(19,45)(22,46)(23,47)(24,48)(26,50)(27,32)(28,53)(30,54)(31,55)(35,57)(36,58)(38,59)(39,60)(42,61)(43,62)(49,63)(51,64)(52,56);(1,6,7,23)(2,11,12,36)(3,15,16,43)(4,18,19,22)(5,20,21,47)(8,26,27,52)(9,30,31,35)(10,33,34,58)(13,38,39,42)(14,40,41,62)(17,44,45,46)(24,49,28,51)(25,50,32,56)(29,54,55,57)(37,59,60,61)(48,63,53,64)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303029, #72303145, #72303161, …
maximal normal subgroup of	#72303029, #72303145, #72303161, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344827);