#509091

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

719 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)(13,14,16,19)(15,24,25,38)(17,30,28,21)(18,33,29,23)(20,37,35,26)(22,40,36,32)(27,43,34,41)(31,42,44,39);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(14,19)(20,35)(21,30)(22,36)(23,33)(24,38)(39,42)(41,43);(2,4)(13,15,16,25)(14,20,19,35)(17,31,28,44)(18,34,29,27)(21,38,30,24)(22,41,36,43)(23,42,33,39)(26,40,37,32);(6,10)(7,12)(14,21)(17,32)(18,29)(19,30)(20,38)(22,33)(23,36)(24,35)(26,31)(27,34)(28,40)(37,44)(39,41)(42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416287);