#8458455

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

4562 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)(6,7)(9,10)(11,19)(12,18)(13,15)(14,17)(16,20)(21,23)(22,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,2,5,4)(3,7,8,6)(9,11,16,14)(10,17,20,19)(12,21,13,22)(15,23,18,24)(27,38)(28,39)(32,37)(33,40);(1,3,5,8)(2,6,4,7)(9,12,16,13)(10,18,20,15)(11,21,14,22)(17,24,19,23)(25,27,31,38)(26,32,36,37)(28,35,39,29)(30,33,34,40);(1,4,5,2)(3,7,8,6)(9,13,16,12)(10,18,20,15)(11,22,14,21)(17,24,19,23)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,34)(36,40);(9,14,16,11)(10,19,20,17)(12,22,13,21)(15,24,18,23)(25,29,31,35)(26,34,36,30)(27,28,38,39)(32,40,37,33);(9,15,16,18)(10,13,20,12)(11,23,14,24)(17,22,19,21)(25,30,31,34)(26,35,36,29)(27,33,38,40)(28,32,39,37)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=g,e^2=gi,f^2=gi,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, 8365651);