#8726687

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

4428 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8633883);