#147940

order	512
nilpotency class	4
isoclinism	isoclinic class of #13316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1000 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)(10,16,18,29)(12,13)(14,31)(17,21,37,35)(19,20,26,33)(22,23)(24,34)(27,30)(28,32)(36,40,39,38);(2,4)(9,10)(11,21)(12,26)(13,29)(14,33)(15,18)(16,32)(17,23)(19,28)(20,31)(22,38)(24,36)(25,35)(27,37)(30,40)(34,39);(6,8)(9,11,15,25)(10,17,18,37)(12,27,28,23)(13,30,32,22)(14,34,31,24)(16,35,29,21)(19,39,26,36)(20,40,33,38)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,d^b=dhi,e^b=ei,f^b=fh,g^b=gi,h^b=hi,d^c=dhi,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, 55136);