#8202409

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

4447 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=h,e^2=g,f^2=g,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, 8109605);