#8362489

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	#36032	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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,e^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,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, 8269685);