#8720551

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
prime radical	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4245 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627747);