#453873

order	512
nilpotency class	3
isoclinism	isoclinic class of #425083
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2576
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16183
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

531 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)(3,14,18,42)(4,20,24,10)(5,11)(6,12)(7,29)(8,13)(9,17,33,19)(15,41)(16,38,45,62)(21,49)(22,48,53,35)(23,37,30,36)(25,54,46,32)(26,39)(27,55)(28,40)(31,47)(34,44)(43,61)(50,64)(51,58)(52,60,56,59)(57,63);(1,3,8,18)(2,9)(4,21)(5,25)(6,17,7,19)(10,34,20,41)(11,38,40,62)(12,14)(13,33)(15,30)(16,27)(22,51,53,43)(23,44)(24,47)(26,45)(28,46)(29,42)(31,37,49,36)(32,39,54,55)(35,57)(48,64)(50,52,63,56)(58,60)(59,61);(1,4,6,23,8,24,7,30)(2,10,12,36,13,20,29,37)(3,15,17,21,18,44,19,47)(5,22,26,52,28,53,27,56)(9,31,14,41,33,49,42,34)(11,35,39,59,40,48,55,60)(16,43,25,50,45,51,46,63)(32,57,38,61,54,64,62,58);(1,5,6,26,8,28,7,27)(2,11,29,55,13,40,12,39)(3,16,19,46,18,45,17,25)(4,22,23,52,24,53,30,56)(9,32,14,38,33,54,42,62)(10,35,37,60,20,48,36,59)(15,43,47,63,44,51,21,50)(31,57,41,61,49,64,34,58)

Polycyclic presentation:

2^9|a^2=efg,b^2=egh,c^2=e,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,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, 361069);