#8147834

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	#40493	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4440 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8055030);