#8254732

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

4347 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)(11,12)(13,21)(14,15)(16,23)(17,20)(18,24)(19,22)(25,26,32,36)(27,37,39,33)(28,30,29,34)(31,35,38,40);(1,2,5,4)(3,7,8,6)(9,11,14,17)(10,12,15,20)(13,22,23,18)(16,24,21,19)(25,27)(26,33)(28,35)(29,40)(30,31)(32,39)(34,38)(36,37);(2,4)(6,7)(10,15)(11,17)(16,21)(18,22)(25,28,32,29)(26,34,36,30)(27,35,39,40)(31,33,38,37);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,22)(23,24)(25,29,32,28)(26,30,36,34)(27,35,39,40)(31,33,38,37);(1,3,5,8)(2,6,4,7)(9,13,14,23)(10,16,15,21)(11,18,17,22)(12,19,20,24)(25,30)(26,28)(27,38)(29,36)(31,39)(32,34)(33,40)(35,37);(1,4,5,2)(3,7,8,6)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,31,32,38)(26,35,36,40)(27,30,39,34)(28,37,29,33)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8161928);