#8689217

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4267 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=g,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, 8596413);