#512467

order	512
nilpotency class	4
isoclinism	isoclinic class of #511954
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

667 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,20,26)(15,27,31,22)(16,21,34,42)(17,35,37,24)(18,38,28,23)(19,39,36,25)(29,41,33,43)(30,40,32,44);(6,9)(10,12)(13,15,18,29,19,30,17,16,20,31,28,33,36,32,37,34)(14,21,24,40,25,41,23,27,26,42,35,44,39,43,38,22);(1,2,3,4)(13,15,28,33,19,30,37,34,20,31,18,29,36,32,17,16)(14,21,35,44,25,41,38,22,26,42,24,40,39,43,23,27);(5,7,8,11)(6,10,9,12)(13,16,17,32,36,29,18,31,20,34,37,30,19,33,28,15)(14,22,23,41,39,44,24,21,26,27,38,43,25,40,35,42)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fh,d^2=e,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=ghi,h^a=hi,c^b=ci,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419663);