#379980

order	512
nilpotency class	3
isoclinism	isoclinic class of #196101
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1460 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)(9,10,15,20,13,18,14,19)(11,21,24,36,23,16,30,32)(12,25,29,34,27,17,28,35)(22,31,26,33,38,37,39,40);(2,5)(6,8)(9,11)(10,16)(12,26)(13,23)(14,30)(15,24)(17,33)(18,21)(19,36)(20,32)(22,28)(25,40)(27,39)(29,38)(31,34)(35,37);(3,7)(6,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,31)(18,25)(19,35)(20,34)(21,37)(23,38)(24,39)(26,30)(32,40)(33,36);(1,3,4,7)(2,6,5,8)(12,27)(17,25)(22,38)(26,39)(28,29)(31,37)(33,40)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287176);