#8691860

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

4567 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)(3,4)(5,7)(6,8)(10,15)(11,12)(18,21)(19,20)(25,26,31,35)(27,32,36,38)(28,33,37,40)(29,30,34,39);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(27,36)(28,37)(32,38)(33,40);(9,11,14,12)(10,16,15,17)(13,19,22,20)(18,23,21,24)(25,27)(26,32)(28,29)(30,33)(31,36)(34,37)(35,38)(39,40);(2,7)(3,8)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,24)(20,23)(25,28)(26,33)(27,29)(30,32)(31,37)(34,36)(35,40)(38,39);(1,4,5,6)(2,3,7,8)(9,12,14,11)(10,17,15,16)(13,19,22,20)(18,23,21,24)(25,29)(26,30)(27,28)(31,34)(32,33)(35,39)(36,37)(38,40);(1,3)(2,4)(5,8)(6,7)(9,13,14,22)(10,18,15,21)(11,19,12,20)(16,23,17,24)(25,30)(26,34)(27,33)(28,38)(29,35)(31,39)(32,37)(36,40)

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^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, 8599056);