#8691801

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

4488 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)(3,7)(4,6)(5,8)(9,10,15,19)(11,20,22,16)(12,21,24,17)(13,14,18,23)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,39)(37,40);(1,2,4,6)(3,5,8,7)(9,11,15,22)(10,16,19,20)(12,23,24,14)(13,21,18,17)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,38)(39,40);(1,2)(3,5)(4,6)(7,8)(25,27)(26,29)(28,35)(30,33)(31,34)(32,39)(36,37)(38,40);(1,3,4,8)(2,5,6,7)(9,12,15,24)(10,17,19,21)(11,14,22,23)(13,16,18,20)(25,27)(26,31)(28,35)(29,34)(30,33)(32,37)(36,39)(38,40);(9,13)(10,14)(11,21)(12,16)(15,18)(17,22)(19,23)(20,24)(25,28)(26,32)(27,35)(29,39)(30,38)(31,37)(33,40)(34,36);(9,14)(10,18)(11,12)(13,19)(15,23)(16,17)(20,21)(22,24)(25,29,30,31)(26,33,34,27)(28,37,38,39)(32,35,36,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8598997);