#9057894

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

4411 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,e^2=h,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, 8965090);