#8460673

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

4421 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,e^2=gi,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, 8367869);