#475188

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
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

681 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)(5,6,12,15,11,16,10,26)(7,17,9,24,20,33,19,13)(8,14,27,30,23,31,22,32)(18,29,34,28,21,35,25,36);(5,7,12,9,11,20,10,19)(6,13,26,33,16,24,15,17)(8,21,22,34,23,18,27,25)(14,28,30,35,31,36,32,29);(2,4)(5,8)(6,14)(7,18)(9,25)(10,27)(11,23)(12,22)(13,28)(15,32)(16,31)(17,29)(19,34)(20,21)(24,36)(26,30)(33,35);(5,9,12,20,11,19,10,7)(6,13,15,17,16,24,26,33)(8,18,27,34,23,21,22,25)(14,29,30,28,31,35,32,36)

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^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, 382384);