#8703517

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

4499 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gh,e^2=h,f^2=g,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, 8610713);