#8665253

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

4506 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=i,e^2=g,f^2=gh,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, 8572449);