#8947191

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	#35698	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4274 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,f^2=g,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, 8854387);