#472330

order	512
nilpotency class	3
isoclinism	isoclinic class of #465214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

912 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)(3,6)(4,5)(7,8)(10,16)(11,21)(12,25)(13,14)(17,31)(19,20)(23,24)(26,37)(27,28)(30,36)(32,34)(39,40);(2,5)(6,8)(9,10,13,16,15,19,14,20)(11,17,23,29,24,31,21,33)(12,18,27,30,28,35,25,36)(22,32,37,39,38,40,26,34);(3,7)(6,8)(9,11,14,21,15,24,13,23)(10,17,16,29,19,31,20,33)(12,26,25,38,28,37,27,22)(18,34,30,32,35,39,36,40);(1,3,4,7)(2,6,5,8)(9,12,13,27,15,28,14,25)(10,18,16,30,19,35,20,36)(11,22,21,26,24,38,23,37)(17,32,33,34,31,40,29,39)

Polycyclic presentation:

2^9|b^2=gh,c^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=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, 379526);