#472224

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

983 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:

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

Polycyclic presentation:

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

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, 379420);