#8346710

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	#34609	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4420 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)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(27,35)(29,39)(31,36)(33,40);(1,2)(3,7)(4,5)(6,8)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,26)(27,36)(28,38)(29,40)(30,34)(31,35)(32,37)(33,39);(2,5)(6,7)(9,11,14,12)(10,16,18,15)(13,20,21,19)(17,22,24,23)(25,27,30,35)(26,31,34,36)(28,29,37,39)(32,40,38,33);(3,8)(6,7)(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22)(25,28)(26,32)(27,29)(30,37)(31,40)(33,36)(34,38)(35,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,29)(26,33)(27,37)(28,35)(30,39)(31,32)(34,40)(36,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)(25,26)(27,36)(28,32)(29,33)(30,34)(31,35)(37,38)(39,40)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=g,f^2=g,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, 8253906);