#441392

order	512
nilpotency class	3
isoclinism	isoclinic class of #418535
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

832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=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, 348588);