#8260810

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

4724 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)(3,5,7,8)(9,10,14,15)(11,17,19,12)(13,16,21,22)(18,23,24,20)(26,28)(31,34)(32,36)(38,40);(1,3,4,7)(2,5,6,8)(10,15)(12,17)(16,22)(20,23)(26,28)(27,33)(32,36)(35,39);(1,2)(3,5)(4,6)(7,8)(13,21)(16,22)(18,24)(20,23)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(9,13,14,21)(10,16,15,22)(11,18,19,24)(12,20,17,23)(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=hi,c^2=i,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8168006);