#8695329

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

4361 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)(3,8,5,4)(10,15)(11,12)(18,21)(19,20);(1,3,6,5)(2,4,7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(26,31)(27,33)(32,39)(34,40);(9,11,14,12)(10,16,15,17)(13,19,22,20)(18,23,21,24)(25,26,30,31)(27,29,33,38)(28,32,36,39)(34,37,40,35);(2,7)(3,5)(9,12,14,11)(10,17,15,16)(13,19,22,20)(18,23,21,24)(25,27)(26,29)(28,34)(30,33)(31,38)(32,37)(35,39)(36,40);(1,4,6,8)(2,3,7,5)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,24)(20,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,39)(33,40)(35,38);(1,5,6,3)(2,4,7,8)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,29)(26,33)(27,31)(28,35)(30,38)(32,34)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8602525);