#8402331

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4399 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,35)(27,34,29,32)(28,33,38,30)(36,40,39,37);(1,2,4,6)(3,5,8,7)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(27,29)(32,34)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(12,20)(15,22)(18,24)(21,23)(25,27,31,29)(26,32,35,34)(28,36,38,39)(30,40,33,37);(1,3)(2,5)(4,8)(6,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,28)(26,33)(27,36)(29,39)(30,35)(31,38)(32,37)(34,40);(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,29,31,27)(26,34,35,32)(28,36,38,39)(30,40,33,37);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,30,31,33)(26,28,35,38)(27,37,29,40)(32,39,34,36)

Polycyclic presentation:

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

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, 8309527);