#8437843

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

4728 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=g,e^2=i,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, 8345039);