#148402

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,f^b=fi,d^c=dh,f^c=fh,h^c=hi

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, 55598);