#411393

order	512
nilpotency class	3
isoclinism	isoclinic class of #411392
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15115
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

694 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318589);