#422131

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1041 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,g^c=gi,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, 329327);