#9249501

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

4750 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=g,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, 9156697);