#9267033

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

4220 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=i,d^2=gi,e^2=gh,f^2=g,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, 9174229);