#8141441

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

4705 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)(3,7)(4,5)(6,8)(9,10)(11,18)(12,16)(13,15)(14,21)(17,22)(19,23)(20,24)(29,37)(32,39)(35,36)(38,40);(2,5)(6,7)(11,12)(16,18)(19,20)(23,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,13,18)(14,19,22,20)(17,23,21,24)(25,27,30,28)(26,31,33,34)(29,35,37,36)(32,38,39,40);(1,3,4,8)(2,6,5,7)(9,12,15,11)(10,16,13,18)(14,20,22,19)(17,23,21,24)(25,28,30,27)(26,31,33,34)(29,36,37,35)(32,38,39,40);(9,13,15,10)(11,18,12,16)(14,21,22,17)(19,24,20,23)(25,26)(27,31)(28,34)(29,32)(30,33)(35,38)(36,40)(37,39);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|c^2=gi,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8048637);