#8750119

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

4425 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=hi,d^2=g,e^2=h,f^2=gi,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, 8657315);