#9174332

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

4448 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9081528);