#8180437

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

4439 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,e^2=h,f^2=gh,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, 8087633);