#9243515

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

4245 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=gh,e^2=g,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, 9150711);