#504457

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ 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)(7,10)(8,9)(11,12)(13,14,16,22,30,38,41,21,17,20,31,37,18,23,29,36)(15,24,25,40,32,44,43,33,26,34,42,35,27,39,28,19);(1,2)(3,4)(13,15,17,26)(14,19,20,33)(16,28,31,43)(18,32,30,27)(21,34,36,24)(22,39,37,44)(23,40,38,35)(25,41,42,29);(2,4)(6,9)(10,12)(14,20)(19,33)(21,36)(22,37)(23,38)(24,34)(35,40)(39,44);(5,7,8,11)(6,10,9,12)(14,21)(16,29)(18,30)(19,34)(20,36)(22,38)(23,37)(24,33)(25,28)(27,32)(31,41)(35,39)(40,44)(42,43)

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=fi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,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, 411653);