#8798714

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

4687 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8705910);