#8390932

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	#59160	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4484 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8298128);