#460863

order	512
nilpotency class	3
isoclinism	isoclinic class of #460861
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

1020 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 368059);