#8662972

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

4616 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)(10,15)(11,19)(12,18)(21,22)(25,26)(27,31)(28,37)(29,32)(30,34)(33,36)(35,40)(38,39);(1,3,4,8)(2,5,6,7)(9,10,14,15)(11,16,19,17)(12,13,18,23)(20,21,24,22)(25,27)(26,31)(28,38)(29,35)(30,36)(32,40)(33,34)(37,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,21)(14,19)(15,17)(18,24)(22,23)(25,28,30,29)(26,32,34,37)(27,35,36,38)(31,39,33,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,20)(13,22)(14,19)(15,16)(18,24)(21,23)(25,26)(27,33)(28,32)(29,37)(30,34)(31,36)(35,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,13)(11,20)(14,18)(15,23)(16,21)(17,22)(19,24)(25,27)(26,33)(28,35)(29,38)(30,36)(31,34)(32,40)(37,39);(9,13)(10,18)(11,21)(12,15)(14,23)(16,24)(17,20)(19,22)(25,29,30,28)(26,32,34,37)(27,35,36,38)(31,40,33,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570168);