#8688237

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

4388 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)(9,10,14,13)(11,16,19,15)(12,17,20,21)(18,23,24,22)(28,36)(31,39)(35,40)(37,38);(3,7)(6,8)(9,11)(10,15)(12,18)(13,16)(14,19)(17,22)(20,24)(21,23)(25,26)(27,32)(28,31)(29,33)(30,34)(35,37)(36,39)(38,40);(1,2)(3,6)(4,5)(7,8)(25,27)(26,29)(28,35)(30,33)(31,38)(32,34)(36,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,18)(13,15)(14,19)(17,23)(20,24)(21,22)(25,28,30,36)(26,31,34,39)(27,35,33,40)(29,38,32,37);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,17,13,21)(11,18,19,24)(15,22,16,23)(25,27)(26,32)(28,35)(29,34)(30,33)(31,37)(36,40)(38,39);(9,13,14,10)(11,16,19,15)(12,17,20,21)(18,22,24,23)(25,29,30,32)(26,33,34,27)(28,37,36,38)(31,35,39,40)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595433);