#9034425

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

4782 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:

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

Polycyclic presentation:

2^9|b^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8941621);