#8826308

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8733504);