#435460

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

971 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)(5,6,12,16,11,15,10,23)(7,17,20,31,21,13,27,32)(8,22,25,26,24,28,14,9)(18,34,33,19,35,36,30,29);(5,7,10,20,11,21,12,27)(6,13,23,32,15,17,16,31)(8,18,25,30,24,35,14,33)(9,19,22,34,26,29,28,36);(5,8,12,25,11,24,10,14)(6,9,23,28,15,26,16,22)(7,18,27,30,21,35,20,33)(13,29,32,34,17,19,31,36);(2,4)(5,9)(6,14)(7,19)(8,23)(10,22)(11,26)(12,28)(13,30)(15,25)(16,24)(17,33)(18,31)(20,34)(21,29)(27,36)(32,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,c^2=fi,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,g^b=gi,h^b=hi,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, 342656);