#450595

order	512
nilpotency class	3
isoclinism	isoclinic class of #417669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15972
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

806 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)(3,15,20,41)(4,21)(5,27,30,42)(6,12)(7,13)(8,14)(9,18,34,19)(10,25)(11,29,40,31)(16,43,45,35)(17,33)(22,50,48,32)(23,52,55,57)(24,38)(26,37)(28,39)(36,56,49,54)(44,58)(46,59)(47,60)(51,62)(53,64)(61,63);(1,3,8,18,7,20,6,19)(2,9,14,15,13,34,12,41)(4,22,26,45,25,48,24,16)(5,28,31,47,30,46,29,17)(10,35,38,32,21,43,37,50)(11,39,42,60,40,59,27,33)(23,53,56,63,55,44,54,51)(36,58,52,62,49,64,57,61);(1,4,8,26,7,25,6,24)(2,10,14,38,13,21,12,37)(3,16,18,22,20,45,19,48)(5,23,31,56,30,55,29,54)(9,32,15,43,34,50,41,35)(11,36,42,52,40,49,27,57)(17,44,28,51,47,53,46,63)(33,58,39,62,60,64,59,61);(1,5,6,29,7,30,8,31)(2,11,12,27,13,40,14,42)(3,17,18,28,20,47,19,46)(4,23,24,54,25,55,26,56)(9,33,15,39,34,60,41,59)(10,36,37,57,21,49,38,52)(16,44,22,51,45,53,48,63)(32,58,43,62,50,64,35,61)

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,c^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,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, 357791);