#504159

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×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

589 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)(13,14,18,23)(15,24,25,35)(16,20,31,39)(17,32,30,22)(19,34,37,26)(21,40,38,29)(27,44,33,41)(28,42,43,36);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,25)(14,19,23,37)(16,28,31,43)(17,33,30,27)(20,24,39,35)(21,41,38,44)(22,42,32,36)(26,40,34,29);(2,4)(5,8,9,7)(6,10,11,12);(14,20)(15,25)(16,29)(17,30)(19,35)(21,22)(23,39)(24,37)(26,43)(28,34)(31,40)(32,38)(36,41)(42,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=f,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, 411355);