#8549667

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

4476 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8456863);