#8361282

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	#47874	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=g,e^2=i,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, 8268478);