#441395

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

851 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)(5,6,10,15,11,16,9,12)(7,17,19,13,20,26,25,28)(8,21,24,29,23,30,22,14)(18,32,33,36,35,27,34,31);(6,12,16,15)(8,22,23,24)(13,26,28,17)(14,29)(18,33,35,34)(21,30)(27,32)(31,36);(5,7,10,19,11,20,9,25)(6,13,12,17,16,28,15,26)(8,18,22,34,23,35,24,33)(14,27,21,31,29,32,30,36);(2,4)(5,8)(6,14)(7,18)(9,24)(10,22)(11,23)(12,21)(13,27)(15,30)(16,29)(17,31)(19,34)(20,35)(25,33)(26,36)(28,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=f,e^2=i,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=de,e^c=ei,e^d=ei,g^d=gi,h^d=hi

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, 348591);