#8341607

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4589 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8248803);