#476845

order	512
nilpotency class	3
isoclinism	isoclinic class of #476095
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	#2684
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

902 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:

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

Polycyclic presentation:

2^9|a^2=ef,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,d^b=def,f^b=fi,d^c=de,f^d=fi

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, 384041);