#8679862

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

4487 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)(9,10,14,15)(11,12,19,16)(13,17,23,24)(18,20,22,21)(25,26,32,29)(27,33,30,28)(31,34,39,37)(35,40,38,36);(3,7)(6,8)(10,15)(12,16)(17,24)(20,21)(25,27,32,30)(26,33,29,28)(31,35,39,38)(34,40,37,36);(1,2,5,4)(3,6,7,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,21)(20,24)(22,23)(25,28)(26,30)(27,29)(31,36)(32,33)(34,38)(35,37)(39,40);(1,3,5,7)(2,6,4,8)(9,12,14,16)(10,11,15,19)(13,21,23,20)(17,22,24,18)(25,29,32,26)(27,33,30,28)(31,34,39,37)(35,36,38,40);(1,4,5,2)(3,8,7,6)(9,11)(10,16)(12,15)(13,22)(14,19)(17,20)(18,23)(21,24)(25,30,32,27)(26,33,29,28)(31,35,39,38)(34,36,37,40);(9,13,14,23)(10,17,15,24)(11,18,19,22)(12,20,16,21)(25,31,32,39)(26,34,29,37)(27,35,30,38)(28,36,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8587058);