#442153

order	512
nilpotency class	3
isoclinism	isoclinic class of #421903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

969 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^c=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, 349349);