#8436877

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

4295 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,e^2=gh,f^2=g,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, 8344073);