#8947351

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4393 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8854547);