#461214

order	512
nilpotency class	3
isoclinism	isoclinic class of #460861
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

901 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)(5,6,10,15,9,16,11,14)(7,12,20,27,18,29,19,28)(8,21,24,31,23,32,25,13)(17,33,35,26,22,36,34,30);(5,7,10,19,9,18,11,20)(6,12,15,28,16,29,14,27)(8,22,24,35,23,17,25,34)(13,30,21,36,31,26,32,33);(5,8,9,23)(6,13,16,31)(7,17,18,22)(10,25,11,24)(12,26,29,30)(14,21,15,32)(19,35,20,34)(27,33,28,36);(2,4)(6,14,16,15)(8,23)(12,27,29,28)(13,21,31,32)(17,22)(24,25)(26,33,30,36)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368410);