#8920744

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

4354 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)(3,5)(4,8)(6,7)(9,10,15,17)(11,16,21,23)(12,18,22,24)(13,19,20,14)(25,26,31,29)(27,32,34,30)(28,33,36,38)(35,39,40,37);(2,7)(3,8)(9,11)(10,16)(12,20)(13,22)(14,18)(15,21)(17,23)(19,24)(25,27,31,34)(26,32,29,30)(28,35,36,40)(33,39,38,37);(1,3,6,8)(2,4,7,5)(10,17)(11,21)(12,22)(14,19)(27,34)(28,36)(30,32)(33,38);(1,4,6,5)(2,3,7,8)(9,12,15,22)(10,18,17,24)(11,20,21,13)(14,23,19,16)(25,28)(26,33)(27,35)(29,38)(30,37)(31,36)(32,39)(34,40);(1,2)(3,4)(5,8)(6,7)(9,13,15,20)(10,19,17,14)(11,12,21,22)(16,18,23,24)(25,29,31,26)(27,30,34,32)(28,33,36,38)(35,39,40,37);(1,5,6,4)(2,8,7,3)(9,14,15,19)(10,20,17,13)(11,18,21,24)(12,23,22,16)(25,30)(26,27)(28,37)(29,34)(31,32)(33,35)(36,39)(38,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8827940);