#8914394

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

4274 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:

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

Polycyclic presentation:

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

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, 8821590);