#506486

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

831 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)(8,10)(9,12)(13,14,17,21,30,20,35,36,18,22,32,38,16,23,29,39)(15,24,27,40,26,43,44,33,25,41,42,34,28,37,31,19);(1,2,3,4)(5,7,8,12)(6,9,10,11)(15,25)(19,33)(24,41)(26,28)(27,42)(31,44)(34,40)(37,43);(2,4)(13,15)(14,19)(16,26)(17,31)(18,25)(20,34)(21,37)(22,33)(23,40)(24,39)(27,29)(28,30)(32,44)(35,42)(36,41)(38,43);(6,10)(9,11)(13,16,18,30)(14,20,22,23)(15,26,25,28)(17,29,32,35)(19,34,33,40)(21,36,38,39)(24,37,41,43)(27,44,42,31)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413682);