#8271161

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

4336 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,d^2=hi,e^2=hi,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, 8178357);