#198719

order	512
nilpotency class	3
isoclinism	isoclinic class of #198498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1324 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105915);