#148563

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

738 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)(9,10,15,21,16,23,31,22)(11,24,28,32,29,39,40,36)(12,20,14,19,13,17,30,33)(18,34,37,26,38,25,35,27);(5,6)(7,8)(10,17)(11,25)(12,30)(13,14)(15,31)(18,35)(19,21)(20,23)(22,33)(26,28)(27,40)(29,34)(32,36)(37,38);(2,4)(6,8)(9,11,12,26,15,28,14,25,16,29,13,27,31,40,30,34)(10,18,19,24,22,35,20,36,23,38,33,39,21,37,17,32)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,h^a=hi,c^b=cf,d^b=dhi,e^b=ehi,f^b=fh,g^b=gi,h^b=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, 55759);