#475187

order	512
nilpotency class	3
isoclinism	isoclinic class of #459486
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

691 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)(5,6,12,16,11,17,10,26)(7,18,25,15,20,33,9,13)(8,14,27,31,23,32,22,34)(19,35,24,28,21,30,36,29);(5,7,12,25,11,20,10,9)(6,13,26,33,17,15,16,18)(8,19,22,36,23,21,27,24)(14,29,31,35,32,28,34,30);(2,4)(5,8)(6,14)(7,19)(9,24)(10,27)(11,23)(12,22)(13,28)(15,29)(16,34)(17,32)(18,35)(20,21)(25,36)(26,31)(30,33);(5,9,10,20,11,25,12,7)(6,15,26,18,17,13,16,33)(8,21,22,24,23,19,27,36)(14,30,34,28,32,35,31,29)

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,h^c=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, 382383);