#9057746

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

4668 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8964942);