#8339877

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	#37620	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 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)(3,4)(5,6)(7,8)(9,10,15,12)(11,16,14,18)(13,17,22,20)(19,23,21,24)(26,28)(27,33)(32,36)(35,39);(2,6)(3,7)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,26,30,28)(27,34,33,31)(29,32,37,36)(35,40,39,38);(1,3,5,7)(2,4,6,8)(9,10,15,12)(11,18,14,16)(13,17,22,20)(19,24,21,23)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38);(1,4)(2,7)(3,6)(5,8)(9,12,15,10)(11,18,14,16)(13,17,22,20)(19,23,21,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,28,30,26)(27,31,33,34)(29,36,37,32)(35,38,39,40);(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8247073);