#9062155

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

4554 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)(3,6)(4,5)(7,8)(9,10,15,18)(11,16,19,14)(12,13,17,24)(20,21,22,23)(25,26,31,29)(27,34,33,30)(28,37,39,32)(35,40,36,38);(1,3)(2,6)(4,8)(5,7)(9,11,15,19)(10,16,18,14)(12,22,17,20)(13,23,24,21)(25,27)(26,30)(28,35)(29,34)(31,33)(32,40)(36,39)(37,38);(2,4)(3,7)(11,19)(12,17)(13,24)(14,16)(25,28,31,39)(26,32,29,37)(27,35,33,36)(30,40,34,38);(1,3,5,7)(2,6,4,8)(9,12)(10,13)(11,20)(14,23)(15,17)(16,21)(18,24)(19,22)(25,28)(26,32)(27,36)(29,37)(30,38)(31,39)(33,35)(34,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,17)(11,21)(12,18)(14,20)(15,24)(16,22)(19,23)(25,29,31,26)(27,34,33,30)(28,32,39,37)(35,40,36,38);(1,4,5,2)(3,8,7,6)(9,14)(10,11)(12,23)(13,20)(15,16)(17,21)(18,19)(22,24)(25,30,31,34)(26,33,29,27)(28,38,39,40)(32,35,37,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8969351);