#463659

order	512
nilpotency class	3
isoclinism	isoclinic class of #455393
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

777 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)(3,15,19,9)(4,20,24,41)(5,25)(6,13)(7,14)(8,12)(10,23,36,30)(11,28)(16,34)(17,32,48,43)(18,40,29,33)(21,42)(22,49,52,61)(26,39)(27,38)(31,45)(35,51,37,54)(44,59)(46,56)(47,58,53,62)(50,57)(55,63)(60,64);(1,3)(2,9)(4,21)(5,17)(6,18)(7,29)(8,19)(10,34)(11,32)(12,15)(13,33)(14,40)(16,23)(20,31)(22,50)(24,45)(25,43)(26,47)(27,53)(28,48)(30,46)(35,59)(36,56)(37,60)(38,58)(39,62)(41,42)(44,51)(49,55)(52,63)(54,64)(57,61);(1,4,6,23,8,24,7,30)(2,10,14,41,12,36,13,20)(3,16,18,45,19,46,29,21)(5,22,26,51,28,52,27,54)(9,31,40,34,15,42,33,56)(11,37,39,49,25,35,38,61)(17,44,47,63,48,64,53,50)(32,57,62,60,43,55,58,59);(1,5,6,26,8,28,7,27)(2,11,14,39,12,25,13,38)(3,17,18,47,19,48,29,53)(4,22,23,51,24,52,30,54)(9,32,40,62,15,43,33,58)(10,35,41,61,36,37,20,49)(16,44,45,63,46,64,21,50)(31,55,34,59,42,57,56,60)

Polycyclic presentation:

2^9|a^2=fg,c^2=ef,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^c=hi,h^d=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, 370855);