#9040768

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

4433 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947964);