#452538

order	512
nilpotency class	3
isoclinism	isoclinic class of #452537
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16051
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=f,c^2=ef,d^2=efi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359734);