#8096217

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

4375 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003413);