#8550944

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

4359 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8458140);