#8402312

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

4452 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309508);