#8688295

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

4567 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)(9,10,14,18)(11,16,12,15)(13,17,22,21)(19,24,20,23)(25,26)(27,35)(28,37)(29,33)(30,31)(32,34)(36,38)(39,40);(1,3,6,4)(2,7,5,8)(11,12)(15,16)(19,20)(23,24)(25,27,32,31)(26,30,34,35)(28,36,29,39)(33,38,37,40);(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,28,32,29)(26,33,34,37)(27,36,31,39)(30,38,35,40);(1,3,6,4)(2,8,5,7)(9,12,14,11)(10,16,18,15)(13,19,22,20)(17,23,21,24)(25,29,32,28)(26,33,34,37)(27,36,31,39)(30,40,35,38);(1,4,6,3)(2,7,5,8)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23)(25,30,32,35)(26,27,34,31)(28,38,29,40)(33,36,37,39);(1,5,6,2)(3,8,4,7)(9,13,14,22)(10,17,18,21)(11,19,12,20)(15,23,16,24)(25,31,32,27)(26,30,34,35)(28,39,29,36)(33,38,37,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gi,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, 8595491);