#476703

order	512
nilpotency class	3
isoclinism	isoclinic class of #476293
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1022 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=e,e^2=i,f^2=i,g^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=def,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=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, 383899);