#455707

order	512
nilpotency class	3
isoclinism	isoclinic class of #454041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1117 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:

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

Polycyclic presentation:

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

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, 362903);