#9034394

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

4618 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,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, 8941590);