#441805

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

827 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)(7,17,20,12)(8,22,24,30)(9,16)(10,14)(11,15)(13,23,31,25)(18,32)(19,27,21,28)(26,33)(29,34)(35,36);(5,7,9,21,11,20,10,19)(6,12,16,27,15,17,14,28)(8,18,25,33,24,35,23,34)(13,29,30,32,31,26,22,36);(5,8,10,25,11,24,9,23)(6,13,16,22,15,31,14,30)(7,18,21,34,20,35,19,33)(12,26,28,36,17,29,27,32);(2,4)(6,14,15,16)(12,27,17,28)(13,30,31,22)(26,32,29,36)

Polycyclic presentation:

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

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, 349001);