#148046

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

788 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)(11,21)(12,17)(13,16)(14,19)(15,23)(18,28)(20,27)(22,30)(24,38)(25,34)(26,32)(29,39)(31,37)(33,40)(35,36);(1,2,3,4)(9,11,12,23)(10,15,17,21)(13,24,26,37)(14,22,27,25)(16,31,32,38)(18,29,33,35)(19,34,20,30)(28,36,40,39);(10,16)(11,22)(13,28)(14,27)(15,29)(17,32)(18,20)(19,33)(21,35)(23,25)(24,37)(26,40)(30,38)(31,34)

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,f^b=fi,e^c=eh,f^c=fi,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, 55242);