#441804

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

859 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)(7,11,17,28)(8,19,22,30)(9,15)(10,13)(12,23,29,34)(14,21)(16,31)(18,27,24,26)(20,33)(25,35)(32,36);(5,7,10,24,9,17,21,18)(6,11,13,26,15,28,14,27)(8,20,34,35,22,16,23,32)(12,25,30,31,29,36,19,33);(5,8,21,34,9,22,10,23)(6,12,13,19,15,29,14,30)(7,16,24,35,17,20,18,32)(11,25,27,31,28,36,26,33);(2,4)(6,13,15,14)(7,17)(11,26,28,27)(12,19,29,30)(16,20)(18,24)(25,33,36,31)(32,35)

Polycyclic presentation:

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

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, 349000);