#504157

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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

605 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411353);