#8899292

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

4600 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,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, 8806488);