#8741959

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

4499 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8649155);