#463660

order	512
nilpotency class	3
isoclinism	isoclinic class of #455397
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16481
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

783 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)(3,15,20,9)(4,21,25,37)(5,26)(6,13)(7,14)(8,12)(10,24,36,30)(11,28)(16,34)(17,47,50,55)(18,41,19,33)(22,42)(23,35,54,38)(27,39)(29,40)(31,45)(32,49,43,48)(44,60)(46,59)(51,57,56,53)(52,62)(58,64)(61,63);(1,3)(2,9)(4,22)(5,17)(6,18)(7,19)(8,20)(10,34)(11,32)(12,15)(13,33)(14,41)(16,24)(21,31)(23,52)(25,45)(26,55)(27,48)(28,49)(29,50)(30,46)(35,61)(36,59)(37,42)(38,62)(39,43)(40,47)(44,53)(51,58)(54,63)(56,60)(57,64);(1,4,6,24,8,25,7,30)(2,10,14,37,12,36,13,21)(3,16,18,45,20,46,19,22)(5,23,27,53,29,54,28,57)(9,31,41,34,15,42,33,59)(11,38,40,51,39,35,26,56)(17,44,48,63,50,64,49,52)(32,60,47,62,43,58,55,61);(1,5,6,27,8,29,7,28)(2,11,14,40,12,39,13,26)(3,17,18,48,20,50,19,49)(4,23,24,53,25,54,30,57)(9,32,41,47,15,43,33,55)(10,35,37,56,36,38,21,51)(16,44,45,63,46,64,22,52)(31,58,34,61,42,60,59,62)

Polycyclic presentation:

2^9|a^2=fg,c^2=ef,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,d^c=de,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, 370856);