#9040990

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948186);