#8461733

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

4493 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)(3,7)(4,5)(6,8)(10,14)(11,17)(16,22)(18,19)(25,26)(27,31)(28,33)(29,34)(30,35)(32,37)(36,38)(39,40);(2,4)(6,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,27,30,37)(26,31,35,32)(28,34,38,40)(29,36,39,33);(1,3)(2,6)(4,7)(5,8)(25,27)(26,32)(28,34)(29,36)(30,37)(31,35)(33,39)(38,40);(1,4,5,2)(3,7,8,6)(9,11,13,17)(10,15,14,21)(12,19,20,18)(16,24,22,23)(25,27)(26,31)(28,34)(29,33)(30,37)(32,35)(36,39)(38,40);(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,28,30,38)(26,33,35,36)(27,34,37,40)(29,32,39,31);(1,3)(2,7)(4,6)(5,8)(9,12,13,20)(10,16,14,22)(11,18,17,19)(15,23,21,24)(25,29)(26,34)(27,36)(28,32)(30,39)(31,38)(33,37)(35,40)

Polycyclic presentation:

2^9|b^2=i,d^2=g,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8368929);