#8258126

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

4601 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)(3,5)(4,7)(6,8)(9,10,14,15)(11,13,18,16)(12,20,21,17)(19,23,24,22)(25,26,31,30)(27,32,34,28)(29,37,39,33)(35,36,38,40);(2,3)(6,7)(10,15)(11,18)(12,21)(22,23)(25,27)(26,32)(28,30)(29,38)(31,34)(33,36)(35,39)(37,40);(1,2,5,3)(4,6,8,7)(10,15)(13,16)(17,20)(22,23)(26,30)(27,34)(33,37)(35,38);(1,3,5,2)(4,7,8,6)(9,11,14,18)(10,16,15,13)(12,19,21,24)(17,23,20,22)(25,28)(26,27)(29,36)(30,34)(31,32)(33,35)(37,38)(39,40);(1,4,5,8)(2,6,3,7)(9,12)(10,17)(11,19)(13,22)(14,21)(15,20)(16,23)(18,24)(25,29,31,39)(26,33,30,37)(27,35,34,38)(28,36,32,40);(9,13,14,16)(10,18,15,11)(12,22,21,23)(17,24,20,19)(25,30,31,26)(27,28,34,32)(29,37,39,33)(35,36,38,40)

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,d^2=g,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, 8165322);