#195418

order	512
nilpotency class	3
isoclinism	isoclinic class of #18376
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1261 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102614);