#9267041

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

4560 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=hi,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9174237);