#8211707

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

4505 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=hi,e^2=h,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, 8118903);