#503519

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

802 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:

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

Polycyclic presentation:

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

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, 410715);