#8402415

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	#45482	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4321 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=i,e^2=g,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, 8309611);