#8718009

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

4620 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=gi,e^2=g,f^2=h,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, 8625205);