#453065

order	512
nilpotency class	3
isoclinism	isoclinic class of #427311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16078
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

679 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:

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

Polycyclic presentation:

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

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, 360261);