#148099

order	512
nilpotency class	4
isoclinism	isoclinic class of #13316
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

724 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,e^c=eh,f^c=fh,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, 55295);