#8180426

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

4538 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8087622);