#503991

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

617 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)(7,11)(8,9)(10,12)(13,14)(15,29)(16,34)(17,36)(18,26)(19,25)(20,40)(21,28)(22,30)(23,33)(24,35)(27,38)(31,44)(32,42)(37,41)(39,43);(1,2)(3,4)(6,9)(10,11)(13,15,19,31,20,32,39,16,21,33,37,35,38,17,18,30)(14,22,26,36,27,24,41,23,28,34,43,42,40,44,25,29);(2,4)(13,16,18,32,38,31,37,15,21,30,39,17,20,35,19,33)(14,23,25,24,40,36,43,22,28,29,41,44,27,42,26,34);(5,7,8,12)(6,10,9,11)(13,17,37,16,20,15,18,35,21,32,19,30,38,33,39,31)(14,24,43,29,27,34,25,36,28,44,26,23,40,22,41,42)

Polycyclic presentation:

2^9|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, 411187);