#504272

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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 ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

548 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=fhi,d^2=ehi,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411468);