#419452

order	512
nilpotency class	3
isoclinism	isoclinic class of #419450
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1397 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi,f^d=fi,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, 326648);