#8188503

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
prime radical	#42894	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4333 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:

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

Polycyclic presentation:

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

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, 8095699);