#8695357

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

4532 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=g,f^2=g,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, 8602553);