#9057751

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

4559 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)(9,10,15,12)(11,17,19,13)(14,20,22,16)(18,21,24,23)(25,26)(27,30)(28,36)(29,33)(31,35)(32,38)(34,40)(37,39);(3,7)(6,8)(9,11,15,19)(10,13,12,17)(14,18,22,24)(16,21,20,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(10,12)(13,17)(14,22)(18,24)(25,27,29,35)(26,31,33,30)(28,37,38,34)(32,39,36,40);(9,12,15,10)(11,17,19,13)(14,20,22,16)(18,23,24,21)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(1,3)(2,6)(4,7)(5,8)(9,13,15,17)(10,11,12,19)(14,21,22,23)(16,18,20,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,14,15,22)(10,16,12,20)(11,18,19,24)(13,21,17,23)(25,28,29,38)(26,32,33,36)(27,34,35,37)(30,39,31,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=g,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, 8964947);