#463919

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371115);