#438191

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

982 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=fi,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,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, 345387);