#501519

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 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)(13,14,18,24,17,23,32,39,19,25,34,42,20,26,35,36)(15,21,28,38,30,41,33,22,16,29,31,43,27,37,44,40);(1,2)(3,4)(6,9)(10,11)(13,15)(14,21)(16,19)(17,30)(18,28)(20,27)(22,39)(23,41)(24,38)(25,29)(26,37)(31,34)(32,33)(35,44)(36,40)(42,43);(2,4)(13,16,19,15)(14,22,25,40)(17,30,20,27)(18,33,34,44)(21,36,29,39)(23,38,26,43)(24,41,42,37)(28,35,31,32);(5,7,8,12)(6,10,9,11)(13,17,19,20)(14,23,25,26)(15,27,16,30)(18,32,34,35)(21,37,29,41)(22,38,40,43)(24,39,42,36)(28,44,31,33)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,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, 408715);