#458875

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,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, 366071);