#8688416

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

4527 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=i,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595612);