#458124

order	512
nilpotency class	3
isoclinism	isoclinic class of #458124
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2731
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16310
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

845 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)(4,21)(5,25)(6,13)(7,12)(8,14)(9,18)(10,23)(11,29)(16,31)(17,47)(19,34)(20,33)(22,38)(24,41)(26,36)(27,39)(28,40)(30,37)(32,50)(35,46)(42,45)(43,51)(44,57)(48,62)(49,54)(52,58)(53,55)(56,61)(59,64)(60,63);(1,3,7,19,8,20,6,18)(2,9,13,33,14,34,12,15)(4,16,23,45,24,46,30,52)(5,17,28,50,29,51,27,49)(10,35,21,42,37,31,41,58)(11,32,40,47,25,54,39,43)(22,44,26,48,53,63,56,64)(36,60,38,62,61,57,55,59);(1,4,7,23,8,24,6,30)(2,10,13,21,14,37,12,41)(3,16,19,45,20,46,18,52)(5,26,28,53,29,56,27,22)(9,31,33,58,34,35,15,42)(11,38,40,61,25,55,39,36)(17,48,50,63,51,64,49,44)(32,59,47,60,54,62,43,57);(1,5,6,27,8,29,7,28)(2,11,12,39,14,25,13,40)(3,17,18,49,20,51,19,50)(4,22,23,26,24,53,30,56)(9,32,15,43,34,54,33,47)(10,36,21,38,37,61,41,55)(16,44,45,48,46,63,52,64)(31,57,58,59,35,60,42,62)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,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, 365320);