#8649014

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	#37748	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,22)(21,24)(26,31)(27,28)(33,38)(35,36);(1,3,4,8)(2,5,6,7)(9,10)(11,17)(12,14)(13,16)(15,19)(18,24)(20,22)(21,23)(25,26,30,31)(27,32,28,34)(29,33,37,38)(35,39,36,40);(1,3)(2,5)(4,8)(6,7)(10,12)(15,17)(16,20)(23,24)(25,27,30,28)(26,32,31,34)(29,35,37,36)(33,39,38,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,17)(13,21)(14,19)(16,24)(18,22)(20,23)(25,28,30,27)(26,32,31,34)(29,35,37,36)(33,40,38,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,17,19,15)(13,20,22,16)(18,24,21,23)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,39)(37,38);(9,13,14,22)(10,16,12,20)(11,18,19,21)(15,23,17,24)(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=g,c^2=hi,d^2=hi,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, 8556210);