#444218

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,6)(3,7,8,5)(9,10,12,13)(11,14,15,16)(17,18,22,28)(19,30,33,25)(20,26,36,21)(23,29,24,41)(27,42,37,39)(31,47,44,43)(32,35,38,48)(34,46,40,45);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19,22,33)(18,25,28,30)(20,35,36,48)(21,38,26,32)(23,40,24,34)(27,31,37,44)(29,46,41,45)(39,47,42,43);(10,13)(14,16)(17,20,24,37,22,36,23,27)(18,26,41,39,28,21,29,42)(19,31,40,48,33,44,34,35)(25,43,46,38,30,47,45,32);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,26)(18,27,28,37)(19,32,33,38)(20,29,36,41)(23,39,24,42)(25,44,30,31)(34,47,40,43)(35,45,48,46)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=gi,d^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,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, 351414);