#8096188

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	#45995	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4319 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)(11,14)(12,21)(13,23)(15,18)(16,19)(17,24)(20,22)(26,32)(28,33)(34,40)(35,38);(1,2)(3,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,19)(16,21)(17,18)(20,24)(22,23)(25,26,31,32)(27,33,37,28)(29,34,36,40)(30,35,39,38);(1,3)(2,5)(4,8)(6,7)(9,12,16,14)(10,11,19,21)(13,22,24,18)(15,17,20,23)(25,27,31,37)(26,33,32,28)(29,30,36,39)(34,35,40,38);(9,13)(10,17)(11,20)(12,22)(14,18)(15,21)(16,24)(19,23)(25,28)(26,27)(29,38)(30,34)(31,33)(32,37)(35,36)(39,40);(1,3)(2,6)(4,8)(5,7)(9,14,16,12)(10,11,19,21)(13,18,24,22)(15,17,20,23)(25,29)(26,34)(27,30)(28,38)(31,36)(32,40)(33,35)(37,39);(1,2)(3,5)(4,7)(6,8)(9,15,16,20)(10,18,19,22)(11,13,21,24)(12,17,14,23)(25,30)(26,35)(27,36)(28,34)(29,37)(31,39)(32,38)(33,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003384);