#426747

order	512
nilpotency class	3
isoclinism	isoclinic class of #417686
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15358
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

994 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)(3,13,17,37)(4,9)(5,21,24,10)(6,11)(7,12)(8,16,30,25)(14,42,45,40)(15,41)(18,32,50,54)(19,33)(20,34)(22,29)(23,38,26,36)(27,39)(28,51,31,44)(35,46)(43,59)(47,57)(48,56)(49,60,52,61)(53,55)(58,64)(62,63);(1,3,6,25,7,17,27,16)(2,8,11,37,12,30,39,13)(4,14,19,51,20,45,53,44)(5,22,26,46,24,47,23,15)(9,31,33,40,34,28,55,42)(10,35,38,57,21,41,36,29)(18,48,52,63,50,64,49,43)(32,59,61,56,54,62,60,58);(1,4,6,19,7,20,27,53)(2,9,11,33,12,34,39,55)(3,14,25,51,17,45,16,44)(5,18,26,52,24,50,23,49)(8,28,37,42,30,31,13,40)(10,32,38,61,21,54,36,60)(15,43,22,48,46,63,47,64)(29,56,35,62,57,58,41,59);(1,5)(2,10)(3,15)(4,18)(6,26)(7,24)(8,29)(9,32)(11,38)(12,21)(13,41)(14,43)(16,47)(17,46)(19,52)(20,50)(22,25)(23,27)(28,56)(30,57)(31,58)(33,61)(34,54)(35,37)(36,39)(40,59)(42,62)(44,64)(45,63)(48,51)(49,53)(55,60)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333943);