#8246181

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	#36440	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4511 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=i,f^2=gi,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, 8153377);