#471077

order	512
nilpotency class	3
isoclinism	isoclinic class of #460970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2759
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16746
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

553 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eh,c^2=e,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,d^c=de,h^c=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, 378273);