#8689861

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

4667 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)(11,19)(12,17)(13,18)(14,15)(16,20)(21,22)(23,24)(25,26)(27,32)(28,35)(29,34)(30,31)(33,38)(36,40)(37,39);(1,2)(3,7)(4,6)(5,8)(10,15)(11,20)(17,22)(18,24)(25,27,31,29)(26,32,30,34)(28,37,38,36)(33,40,35,39);(1,3)(2,5)(4,8)(6,7)(25,28)(26,33)(27,36)(29,37)(30,35)(31,38)(32,39)(34,40);(2,6)(5,7)(9,11)(10,16)(12,18)(13,22)(14,20)(15,19)(17,23)(21,24)(25,29,31,27)(26,34,30,32)(28,37,38,36)(33,40,35,39);(1,3)(2,7)(4,8)(5,6)(9,12)(10,17)(11,18)(13,19)(14,21)(15,22)(16,23)(20,24)(25,30,31,26)(27,34,29,32)(28,35,38,33)(36,40,37,39);(1,2)(3,5)(4,6)(7,8)(9,13,14,23)(10,18,15,24)(11,17,20,22)(12,16,21,19)(25,28)(26,35)(27,37)(29,36)(30,33)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=h,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, 8597057);