#476611

order	512
nilpotency class	3
isoclinism	isoclinic class of #476609
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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1030 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383807);