#9041109

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

4612 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)(9,10,14,16)(11,15,20,17)(12,18,23,22)(13,21,24,19)(25,26)(27,28)(29,31)(30,32)(33,35)(34,36)(37,38)(39,40);(3,7)(5,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,17)(18,24)(19,23)(25,26,30,32)(27,28,33,35)(29,31,37,38)(34,36,39,40);(1,2)(3,5)(4,6)(7,8)(10,16)(11,20)(12,23)(13,24)(27,33)(28,35)(29,37)(31,38);(9,11)(10,17)(12,19)(13,22)(14,20)(15,16)(18,24)(21,23)(25,27,30,33)(26,28,32,35)(29,34,37,39)(31,36,38,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,18)(11,21)(13,17)(14,23)(15,24)(16,22)(19,20)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,13)(10,19)(11,22)(12,17)(14,24)(15,23)(16,21)(18,20)(25,29,30,37)(26,31,32,38)(27,34,33,39)(28,36,35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=g,f^2=g,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, 8948305);