#8291630

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

4510 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)(3,5)(4,7)(6,8)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,26,31,34)(27,32,29,33)(28,37,38,30)(35,36,39,40);(1,3,6,4)(2,7,8,5)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,26)(27,32)(28,30)(29,33)(31,34)(35,40)(36,39)(37,38);(2,8)(5,7)(11,17)(15,18)(19,23)(22,24)(25,27,31,29)(26,32,34,33)(28,35,38,39)(30,40,37,36);(2,8)(3,4)(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,28)(26,30)(27,35)(29,39)(31,38)(32,40)(33,36)(34,37);(1,4,6,3)(2,5,8,7)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29,31,27)(26,33,34,32)(28,35,38,39)(30,40,37,36);(1,5)(2,3)(4,8)(6,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,30)(26,28)(27,36)(29,40)(31,37)(32,39)(33,35)(34,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=i,e^2=g,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, 8198826);