#504267

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×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

572 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)(15,27)(16,33)(17,24)(18,23)(19,37)(20,26)(21,29)(22,35)(25,28)(30,43)(31,41)(32,40)(34,42)(36,39)(38,44);(1,2)(3,4)(6,9)(10,11)(13,15,18,30,19,31,34,44,20,32,36,16,28,35,17,29)(14,21,24,22,25,33,39,40,26,38,42,41,37,43,23,27);(2,4)(13,16,34,15,28,44,18,35,20,30,17,32,19,29,36,31)(14,22,39,38,37,27,24,33,26,41,23,21,25,40,42,43);(5,7,8,12)(6,10,9,11)(13,15,18,30,19,31,34,44,20,32,36,16,28,35,17,29)(14,21,24,22,25,33,39,40,26,38,42,41,37,43,23,27)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=fh,d^2=eh,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, 411463);