#8648906

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8556102);