#148089

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

766 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)(9,10,14,19)(11,16,24,36)(12,17,29,38)(13,30,27,18)(15,31,28,20)(21,39,22,35)(23,40,26,34)(25,33,32,37);(1,2,3,4)(9,11)(10,16)(12,26)(13,21)(14,24)(15,32)(17,34)(18,35)(19,36)(20,33)(22,27)(23,29)(25,28)(30,39)(31,37)(38,40);(10,17)(11,21)(12,27)(13,29)(15,28)(16,33)(18,20)(19,38)(22,24)(23,32)(25,26)(30,31)(35,39)(36,37)

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,e^c=eh,f^c=fhi,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, 55285);