#9062139

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

4516 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969335);