#441393

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fi,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, 348589);