#8963497

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

4304 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8870693);