#433013

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

989 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^c=hi,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, 340209);