#8942955

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

4654 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)(11,12)(13,21)(14,15)(16,19)(17,22)(18,23)(20,24)(25,26,29,32)(27,30,33,34)(28,31,36,38)(35,37,39,40);(1,2)(3,5)(4,6)(7,8)(10,15)(12,16)(17,21)(20,23)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40);(1,2,4,6)(3,7,8,5)(10,15)(12,16)(13,22)(18,24)(27,33)(28,36)(30,34)(31,38);(3,8)(5,7)(9,11,14,19)(10,16,15,12)(13,18,22,24)(17,23,21,20)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(1,3)(2,5)(4,8)(6,7)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,27,29,33)(26,30,32,34)(28,35,36,39)(31,37,38,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)(25,28)(26,31)(27,35)(29,36)(30,37)(32,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=h,e^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, 8850151);