#463895

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

849 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,8)(3,6,4,7)(9,10,15,29,13,18,14,19)(11,16,27,35,22,32,23,33)(12,24,28,34,25,17,26,36)(20,37,40,31,38,30,21,39);(1,3,5,4)(2,6,8,7)(9,11,13,22)(10,16,18,32)(12,21,25,40)(14,27,15,23)(17,31,24,39)(19,35,29,33)(20,28,38,26)(30,36,37,34);(1,3,5,4)(2,7,8,6)(9,12,13,25)(10,17,18,24)(11,20,22,38)(14,28,15,26)(16,30,32,37)(19,36,29,34)(21,23,40,27)(31,33,39,35);(1,4,5,3)(2,7,8,6)(9,12)(10,17,18,24)(11,21)(13,25)(14,26)(15,28)(16,31,32,39)(19,34,29,36)(20,27)(22,40)(23,38)(30,33,37,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371091);