#459490

order	512
nilpotency class	3
isoclinism	isoclinic class of #459490
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2733
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16341
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1018 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366686);